Some useful fractions to learn

we have to calculate 1/x when x is
x    1/x 
2     .5
3     .{3}
4     .25
5     .20
6     .1{6}
7     .{142857}
8     .125
9     .{1}
10   .1
11    .{09}

Mixture of tough and easy words

Verbal
1. Depreciation: deflation, depression, devaluation, fall, slump
2. Deprecate : feel and express disapproval,
3. Incentive : thing one encourages one to do (stimulus)
4. Echelon : level of authority or responsibility
5. Innovation : make changes or introduce new things
6. Intermittent : externally stopping and then starting
7. Detrimental: harmful
8. Conciliation : make less angry or more friendly

9. Orthodox: conventional or traditional, superstitious

10. Fallible : liable to error

11. Volatile : ever changing

12. Manifest: clear and obvious

13. Connotation : suggest or implied meaning of expression

14. Reciprocal: reverse or opposite

15. Agrarian : related to agriculture

16. Vacillate : undecided or dilemma

17. Expedient : fitting proper, desirable

18. Simulate : produce artificially resembling an existing one.

19. Access : to approach

20. Compensation: salary

21. Truncate : shorten by cutting

22. Adherence : stick

23. Heterogeneous: non similar things

24. Surplus : excessive

25. Assess : determine the amount or value

26. Cognizance : knowledge

27. Retrospective : review

28. Naive : innocent, rustic

29. Equivocate : tallying on both sides, lie, mislead

30. Postulate : frame a theory

31. Latent : dormant, secret

32. Fluctuation : wavering,

33. Eliminate : to reduce

34. Affinity : strong liking

35. Expedite : hasten

36. Console : to show sympathy

37. Adversary : opposition

38. Affable : lovable or approachable

39. Decomposition : rotten

40. Agregious : apart from the crowd, especially bad

41. Conglomeration: group, collection

42. Aberration: deviation

43. Augury : prediction

44. Creditability : ability to common belief, quality of being credible

45. Coincident: incidentally

46. Constituent : accompanying

47. Differential : having or showing or making use of

48. Litigation : engaging in a law suit

49. Moratorium: legally or officially determined period of delay before
fulfillment of the agreement of paying of debts.

50. Negotiate : discuss or bargain

51. Preparation : act of preparing

52. Preponderant : superiority of power or quality

53. Relevance : quality of being relevant

54. Apparatus : appliances

55. Ignorance : blindness, in experience

56. Obsession: complex enthusiasm

57. precipitate : speed, active
58. corroborative: refutable
59. obnoxious : harmless
60. sanction: hinder
61. empirical: experimental
62. aborigine: emigrant

Find all sets of consecutive integers that add up to 1000.

There are total 8 such series:
1.
Sum of 2000 numbers starting from -999 i.e. summation of numbers from -999 to 1000. (-999) + (-998) + (-997) + ..... + (-1) + 0 + 1 + 2 + ..... + 997 + 998 + 999 + 1000 = 1000
2.
Sum of 400 numbers starting from -197 i.e. summation of numbers from -197 to 202. (-197) + (-196) + (-195) + ..... + (-1) + 0 + 1 + 2 + ..... + 199 + 200 + 201 + 202 = 1000
3.
Sum of 125 numbers starting from -54 i.e. summation of numbers from -54 to 70. (-54) + (-53) + (-52) + ..... + (-1) + 0 + 1 + 2 + ..... + 68 + 69 + 70 = 1000
4.
Sum of 80 numbers starting from -27 i.e. summation of numbers from -27 to 52. (-27) + (-26) + (-25) + ..... + (-1) + 0 + 1 + 2 + ..... + 50 + 51 + 52 = 1000
5.
Sum of 25 numbers starting from 28 i.e. summation of numbers from 28 to 52. 28 + 29 + 30 + 31 + 32 + 33 + 34 + 35 + 36 + 37 + 38 + 39 + 40 + 41 + 42 + 43 + 44 + 45 + 46 + 47 + 48 + 49 + 50 + 51 + 52 = 1000
6.
Sum of 16 numbers starting from 55 i.e. summation of numbers from 55 to 70. 55 + 56 + 57 + 58 + 59 +60 + 61 + 62 + 63 + 64 + 65 + 66 + 67 + 68 + 69 + 70 = 1000
7.
Sum of 5 numbers starting from 198 i.e. summation of numbers from 198 to 202. 198 + 199 + 200 +201 + 202 = 1000.
8.
Sum of 1 number starting from 1000. 1000 = 1000

The smallest no. such that if its rightmost digit is placed at its left end, the new number so formed is precisely 50% larger than the original no.

Ques Find the smallest number such that if its rightmost digit is placed at its left end, the new number so formed is precisely 50% larger than the original number.
Answer The answer is 285714. If its rightmost digit is placed at its left end, then new number is 428571 which is 50% larger than the original number 285714.

Sum of last digit = sum of rest digits

Consider a number 235, where last digit is the sum of first two digits i.e. 2 + 3 = 5. How many such 3-digit numbers are there?
Ans There are 45 different 3-digit numbers.

Take the combinations
The last digit can not be 0.
If the last digit is 1, the only possible number is 101. (Note that 011 is not a 3-digit number)
If the last digit is 2, the possible numbers are 202 and 112.
If the last digit is 3, the possible numbers are 303, 213 and 123.
If the last digit is 4, the possible numbers are 404, 314, 224 and 134.
If the last digit is 5, the possible numbers are 505, 415, 325, 235 and 145.

Note the pattern here - If the last digit is 1, there is only one number. If the last digit is 2, there are two numbers. If the last digit is 3, there are three numbers. If the last digit is 4, there are four numbers. If the last digit is 5, there are five numbers. And so on.....
Thus, total numbers are 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 = 45

Altogether then, there are 45 different 3-digit numbers, where last digit is the sum of first two digits.

Divisibility tests

A number is divisible by 2 if its last digit is also (i.e. 0,2,4,6 or 8).
A number is divisible by 3 if the sum of its digits is also. Example: 534: 5+3+4=12 and 1+2=3 so 534 is divisible by 3.
A number is divisible by 5 if the last digit is 5 or 0.
Most people know (only) those 3 rules. Here are my rules for divisibility by the PRIMES up to 50. Why only primes and not also composite numbers? A number is divisible by a composite if it is also divisible by all the prime factors (e.g. is divisible by 21 if divisible by 3 AND by 7). Small numbers are used in these worked examples, so you could have used a pocket calculator. But my rules apply to any number of digits, whereas you cannot test a 30 or more digit number on your pocket calculator otherwise.
Test for divisibility by 7. Double the last digit and subtract it from the remaining leading truncated number. If the result is divisible by 7, then so was the original number. Apply this rule over and over again as necessary. Example: 826. Twice 6 is 12. So take 12 from the truncated 82. Now 82-12=70. This is divisible by 7, so 826 is divisible by 7 also.
There are similar rules for the remaining primes under 40, i.e. 11,13, 17,19,23,29,31,37,41,43 and 47.
Test for divisibility by 11. Subtract the last digit from the remaining leading truncated number. If the result is divisible by 11, then so was the first number. Apply this rule over and over again as necessary.
Example: 19151--> 1915-1 =1914 -->191-4=187 -->18-7=11, so yes, 19151 is divisible by 11.
Test for divisibility by 13. Add four times the last digit to the remaining leading truncated number. If the result is divisible by 13, then so was the first number. Apply this rule over and over again as necessary.
Example: 50661-->5066+4=5070-->507+0=507-->50+28=78 and 78 is 6*13, so 50661 is divisible by 13.
Test for divisibility by 17. Subtract five times the last digit from the remaining leading truncated number. If the result is divisible by 17, then so was the first number. Apply this rule over and over again as necessary.
Example: 3978-->397-5*8=357-->35-5*7=0. So 3978 is divisible by 17.
Test for divisibility by 19. Add two times the last digit to the remaining leading truncated number. If the result is divisible by 19, then so was the first number. Apply this rule over and over again as necessary.
EG: 101156-->10115+2*6=10127-->1012+2*7=1026-->102+2*6=114 and 114=6*19, so 101156 is divisible by 19.
My original divisibilty webpage stopped here. However, I have had a number of mails asking for divisibility tests for larger primes, so I've extended the list up to 50. Actually even with 37 most people cannot do the necessary mental arithmetic easily, because they cannot recognise even single-digit multiples of two-digit numbers on sight. People are no longer taught the multiplication table up to 20*20 as I was as a child. Nowadays we are lucky if they know it up to 10*10.
Test for divisibility by 23. Add seven times the last digit to the remaining leading truncated number. If the result is divisible by 23, then so was the first number. Apply this rule over and over again as necessary.
Example: 17043-->1704+7*3=1725-->172+7*5=207-->20+7*7=69 which is 3*23, so 17043 is also divisible by 23.
Test for divisibility by 29. Add three times the last digit to the remaining leading truncated number. If the result is divisible by 29, then so was the first number. Apply this rule over and over again as necessary.
Example: 15689-->1568+3*9=1595-->159+3*5=174-->17+3*4=29, so 15689 is also divisible by 29.
Test for divisibility by 31. Subtract three times the last digit from the remaining leading truncated number. If the result is divisible by 31, then so was the first number. Apply this rule over and over again as necessary.
Example: 7998-->799-3*8=775-->77-3*5=62 which is twice 31, so 7998 is also divisible by 31.
Test for divisibility by 37. This is (slightly) more difficult, since it perforce uses a double-digit multiplier, namely eleven. People can usually do single digit multiples of 11, so we can use the same technique still. Subtract eleven times the last digit from the remaining leading truncated number. If the result is divisible by 37, then so was the first number. Apply this rule over and over again as necessary.
Example: 23384-->2338-11*4=2294-->229-11*4=185 which is five times 37, so 23384 is also divisible by 37.
Test for divisibility by 41. Subtract four times the last digit from the remaining leading truncated number. If the result is divisible by 41, then so was the first number. Apply this rule over and over again as necessary.
Example: 30873-->3087-4*3=3075-->307-4*5=287-->28-4*7=0, remainder is zero and so 30873 is also divisible by 41.
Test for divisibility by 43. Now it starts to get really difficult for most people, because the multiplier to be used is 13, and most people cannot recognise even single digit multiples of 13 at sight. You may want to make a little list of 13*N first. Nevertheless, for the sake of completeness, we will use the same method. Add thirteen times the last digit to the remaining leading truncated number. If the result is divisible by 43, then so was the first number. Apply this rule over and over again as necessary.
Example: 3182-->318+13*2=344-->34+13*4=86 which is recognisably twice 43, and so 3182 is also divisible by 43.
Update : Bill Malloy has pointed out that, since we are working to modulo43, instead of adding factor 13 times the last digit, we can subtract 30 times it, because 13+30=43. Why didn't I think of that!!! :-(
Finally, the Test for divisibility by 47. This too is difficult for most people, because the multiplier to be used is 14, and most people cannot recognise even single digit multiples of 14 at sight. You may want to make a little list of 14*N first. Nevertheless, for the sake of completeness, we will use the same method. Subtract fourteen times the last digit from the remaining leading truncated number. If the result is divisible by 47, then so was the first number. Apply this rule over and over again as necessary.
Example: 34827-->3482-14*7=3384-->338-14*4=282-->28-14*2=0 , remainder is zero and so 34827 is divisible by 47.
I've stopped here at the last prime below 50, for arbitrary but pragmatic reasons as explained above.
Other blogreaders (sadly even people from .edu domains, who should be able to do the elementary algebra themselves) have asked why I sometimes say ADD and for other primes say SUBTRACT, and ask where the apparently arbitrary factors come from. So let us do some algebra to show the method in my madness.
We have displayed the recursive divisibility test of number N as f-M*r where f are the front digits of N, r is the rear digit of N and M is some multiplier. And we want to see if N is divisible by some prime P. We need a method to work out the values of M. What you do is to calculate (mentally) the smallest multiple of P which ends in a 9 or a 1. If it's a 9 we are going to ADD, if it's a 1 we are going to SUBTRACT later. Then we will use the leading digit(s) of the multiple as our multiplier M.
Example for P=17 : three times 17 is 51 which is the smallest multiple of 17 that ends in a 1 or 9. Since it's a 1 we are going to SUBTRACT later. The leading digit is a 5, so we are going to SUBTRACT five times the remainder r. The algorithm was stated above. Now let's do the algebraic proof. Writing N=10f+r, we can multiply by -5 (as shown in the example for 17), getting -5N=-50f-5r. Now we add 51f to both sides (because 51 was the smallest multiple of P=17 to end in a 1 or a 9), giving one f (which we want), so 51f-5N=f-5r. Now if N is divisible by P (here P=17), we can substitute to get 51f-5*17*x=f-5r and rearrange the left side as 17*(3f-5x)=f-5r and therefore f-5r is a multiple of P=17 also. Q.E.D.

Special numbers

A number of 9 digits has the following properties:

• The number comprising the leftmost two digits is divisible by 2, that comprising the leftmost three digits is divisible by 3, the leftmost four by 4, the leftmost five by 5, and so on for the nine digits of the number i.e. the number formed from the first n digits is divisible by n, 2<=n<=9.

• Each digit in the number is different i.e. no digits are repeated.

• The digit 0 does not occur in the number i.e. it is comprised only of the digits 1-9 in some order.

Find the number.

Answer
The answer is 381654729 One way to solve it is Trial-&-Error. You can make it bit easier as odd positions will always occupy ODD numbers and even positions will always occupy EVEN numbers. Further 5th position will contain 5 as 0 does not occur.
At odd points we have  3 1 5 7 9
At even points we have 8 6 4 2

Antonyms

1)      Mollify            × Enrage, Provoke

2)      Inundate          × Drain

3)      Equanimity      × Agitation

4)      Gauche            × Tactful

5)      Exhume           × Inter

6)      Baleful                        × Beneficent

7)      Anathematize  × Bless

8)      Enigmatic        × Unambiguous

9)      Pariah              × Idol

10)  Turbid             × Limpid

Daily Do's and dont(September)

P's
English related(1 hour)

1. Read 2 flash cards(10 min)
2. Read a news paper(30 min)
3. Read a novel (15 min)
4. Re-read 2 flash cards (5 min)
5.

Computer Related(2-3 hour)

1. Study OS 1 hour
2. Study networking 1 hour
3. Project work sometimes

Guitar Related

Puzzle
1. Daily fogcreek puzzle(5 daily)

5 september vocab

precarious - unstable, uncertain, depending on chance, insecure; delicate

precedent - the original of a copy; model; former; previous

precept  - law; an axiom ; an instruction in moral conduct ; principle ; a rule of action

precinct - a division of police or electoral purposes ; a place around a building

preciosity - fastidious over-refinement

precipice -  a high verical or nearly vertical cliff or rock face.

precipitate - to accelerate, to settle down or suspend in solution , a substance separated from solution due to precipitation

preclude - to stop or obstruct

precocious - strikingly advanced or mature in mental development, speech, social behaviour, etc

Divisibility tests

A number is divisible by 2 if its last digit is also (i.e. 0,2,4,6 or 8).

A number is divisible by 3 if the sum of its digits is also. Example: 534: 5+3+4=12 and 1+2=3 so 534 is divisible by 3.

A number is divisible by 5 if the last digit is 5 or 0.
 .
Test for divisibility by 7. Double the last digit and subtract it from the remaining leading truncated number. If the result is divisible by 7, then so was the original number. Apply this rule over and over again as necessary. Example: 826. Twice 6 is 12. So take 12 from the truncated 82. Now 82-12=70. This is divisible by 7, so 826 is divisible by 7 also.

There are similar rules for the remaining primes under 40, i.e. 11,13, 17,19,23,29,31,37,41,43 and 47.
Test for divisibility by 11. Subtract the last digit from the remaining leading truncated number. If the result is divisible by 11, then so was the first number. Apply this rule over and over again as necessary.
Example: 19151--> 1915-1 =1914 -->191-4=187 -->18-7=11, so yes, 19151 is divisible by 11.


Test for divisibility by 13. Add four times the last digit to the remaining leading truncated number. If the result is divisible by 13, then so was the first number. Apply this rule over and over again as necessary.
Example: 50661-->5066+4=5070-->507+0=507-->50+28=78 and 78 is 6*13, so 50661 is divisible by 13.

Test for divisibility by 17. Subtract five times the last digit from the remaining leading truncated number. If the result is divisible by 17, then so was the first number. Apply this rule over and over again as necessary.
Example: 3978-->397-5*8=357-->35-5*7=0. So 3978 is divisible by 17.

Test for divisibility by 19. Add two times the last digit to the remaining leading truncated number. If the result is divisible by 19, then so was the first number. Apply this rule over and over again as necessary.
EG: 101156-->10115+2*6=10127-->1012+2*7=1026-->102+2*6=114 and 114=6*19, so 101156 is divisible by 19.

If you want you can stop here.However, I have extended the list up to 50.

Test for divisibility by 23. Add seven times the last digit to the remaining leading truncated number. If the result is divisible by 23, then so was the first number. Apply this rule over and over again as necessary.
Example: 17043-->1704+7*3=1725-->172+7*5=207-->20+7*7=69 which is 3*23, so 17043 is also divisible by 23.

Test for divisibility by 29. Add three times the last digit to the remaining leading truncated number. If the result is divisible by 29, then so was the first number. Apply this rule over and over again as necessary.
Example: 15689-->1568+3*9=1595-->159+3*5=174-->17+3*4=29, so 15689 is also divisible by 29.

Test for divisibility by 31. Subtract three times the last digit from the remaining leading truncated number. If the result is divisible by 31, then so was the first number. Apply this rule over and over again as necessary.
Example: 7998-->799-3*8=775-->77-3*5=62 which is twice 31, so 7998 is also divisible by 31.

Test for divisibility by 37. This is (slightly) more difficult, since it perforce uses a double-digit multiplier, namely eleven. People can usually do single digit multiples of 11, so we can use the same technique still. Subtract eleven times the last digit from the remaining leading truncated number. If the result is divisible by 37, then so was the first number. Apply this rule over and over again as necessary.
Example: 23384-->2338-11*4=2294-->229-11*4=185 which is five times 37, so 23384 is also divisible by 37.

Test for divisibility by 41. Subtract four times the last digit from the remaining leading truncated number. If the result is divisible by 41, then so was the first number. Apply this rule over and over again as necessary.
Example: 30873-->3087-4*3=3075-->307-4*5=287-->28-4*7=0, remainder is zero and so 30873 is also divisible by 41.

Test for divisibility by 43. Now it starts to get really difficult for most people, because the multiplier to be used is 13, and most people cannot recognise even single digit multiples of 13 at sight. You may want to make a little list of 13*N first. Nevertheless, for the sake of completeness, we will use the same method. Add thirteen times the last digit to the remaining leading truncated number. If the result is divisible by 43, then so was the first number. Apply this rule over and over again as necessary.
Example: 3182-->318+13*2=344-->34+13*4=86 which is recognisably twice 43, and so 3182 is also divisible by 43.
We can also do following:, since we are working to modulo43, instead of adding factor 13 times the last digit, we can subtract 30 times it, because 13+30=43.

Finally, the Test for divisibility by 47. This too is difficult for most people, because the multiplier to be used is 14, and most people cannot recognise even single digit multiples of 14 at sight. You may want to make a little list of 14*N first. Nevertheless, for the sake of completeness, we will use the same method. Subtract fourteen times the last digit from the remaining leading truncated number. If the result is divisible by 47, then so was the first number. Apply this rule over and over again as necessary.
Example: 34827-->3482-14*7=3384-->338-14*4=282-->28-14*2=0 , remainder is zero and so 34827 is divisible by 47.

Why d we sometimes say ADD and for other primes say SUBTRACT, and where from the apparently arbitrary factors come from. So let us do some algebra to show the method in my madness.
We know that recursive divisibility test of number N as f-M*r where f are the front digits of N, r is the rear digit of N and M is some multiplier. And we want to see if N is divisible by some prime P. We need a method to work out the values of M. What you do is to calculate (mentally) the smallest multiple of P which ends in a 9 or a 1. If it's a 9 we are going to ADD, if it's a 1 we are going to SUBTRACT later. Then we will use the leading digit(s) of the multiple as our multiplier M.
Example for P=17 : three times 17 is 51 which is the smallest multiple of 17 that ends in a 1 or 9. Since it's a 1 we are going to SUBTRACT later. The leading digit is a 5, so we are going to SUBTRACT five times the remainder r. The algorithm was stated above. Now let's do the algebraic proof. Writing N=10f+r, we can multiply by -5 (as shown in the example for 17), getting -5N=-50f-5r. Now we add 51f to both sides (because 51 was the smallest multiple of P=17 to end in a 1 or a 9), giving one f (which we want), so 51f-5N=f-5r. Now if N is divisible by P (here P=17), we can substitute to get 51f-5*17*x=f-5r and rearrange the left side as 17*(3f-5x)=f-5r and therefore f-5r is a multiple of P=17 also. Q.E.D.

Typing

5th septeber - 41wpm  97% accuracy(Aesop's puzzle) Lesson 4

Powers

Some useful fractions to learn

we have to calculate 1/x when x is
x    1/x
2     .5
3     .{3}
4     .25
5     .20
6     .1{6}
7     .{142857}
8     .125
9     .{1}
10   .1
11    .{09}

September Guitar practise

Guitar
Singalong
Last Kiss
Time of your life
Wake me up when september ends
Show me the meaning
Sayonee
Allah k bande


Finger Picking
Stairway to heaven
Anouk - michele
Radiohead - street spirit
Babe I m gonna leave u
Nothing else matters
Under the bridge
Hotel California



Solo
Fade to black intro(4 th september)

Time table I want to practise

Eve 5.00 - 7.00 vocab
       7.00 - 8.00 coding
       8.00 - 9.00 dinner etc
       9.00 -12.00cs subject
       12.00-2.00 phone
       2.00-5.00 sleep
       5.00- 7

Quadratic Equation

Maxima and minima
Maxima a>0  x= - b/2a  Max = -(b^2 -4ac)/4a
Minima a<0  -----------------------------------------(same)

Multiplication Tables

1 × 1 = 1	2 × 1 = 2 2 × 2 = 4	3 × 1 = 3 3 × 2 = 6 3 × 3 = 9	4 × 1 = 4 4 × 2 = 8 4 × 3 = 12 4 × 4 = 16	5 × 1 = 5 5 × 2 = 10 5 × 3 = 15 5 × 4 = 20 5 × 5 = 25
6 × 1 = 6 6 × 2 = 12 6 × 3 = 18 6 × 4 = 24 6 × 5 = 30 6 × 6 = 36	7 × 1 = 7 7 × 2 = 14 7 × 3 = 21 7 × 4 = 28 7 × 5 = 35 7 × 6 = 42 7 × 7 = 49	8 × 1 = 8 8 × 2 = 16 8 × 3 = 24 8 × 4 = 32 8 × 5 = 40 8 × 6 = 48 8 × 7 = 56 8 × 8 = 64	9 × 1 = 9 9 × 2 = 18 9 × 3 = 27 9 × 4 = 36 9 × 5 = 45 9 × 6 = 54 9 × 7 = 63 9 × 8 = 72 9 × 9 = 81	10 × 1 = 10 10 × 2 = 20 10 × 3 = 30 10 × 4 = 40 10 × 5 = 50 10 × 6 = 60 10 × 7 = 70 10 × 8 = 80 10 × 9 = 90 10 × 10 = 100
11 × 1 = 11 11 × 2 = 22 11 × 3 = 33 11 × 4 = 44 11 × 5 = 55 11 × 6 = 66 11 × 7 = 77 11 × 8 = 88 11 × 9 = 99 11 × 10 = 110 11 × 11 = 121	12 × 1 = 12 12 × 2 = 24 12 × 3 = 36 12 × 4 = 48 12 × 5 = 60 12 × 6 = 72 12 × 7 = 84 12 × 8 = 96 12 × 9 = 108 12 × 10 = 120 12 × 11 = 132 12 × 12 = 144	13 × 1 = 13 13 × 2 = 26 13 × 3 = 39 13 × 4 = 52 13 × 5 = 65 13 × 6 = 78 13 × 7 = 91 13 × 8 = 104 13 × 9 = 117 13 × 10 = 130 13 × 11 = 143 13 × 12 = 156 13 × 13 = 169	14 × 1 = 14 14 × 2 = 28 14 × 3 = 42 14 × 4 = 56 14 × 5 = 70 14 × 6 = 84 14 × 7 = 98 14 × 8 = 112 14 × 9 = 126 14 × 10 = 140 14 × 11 = 154 14 × 12 = 168 14 × 13 = 182 14 × 14 = 196	15 × 1 = 15 15 × 2 = 30 15 × 3 = 45 15 × 4 = 60 15 × 5 = 75 15 × 6 = 90 15 × 7 = 105 15 × 8 = 120 15 × 9 = 135 15 × 10 = 150 15 × 11 = 165 15 × 12 = 180 15 × 13 = 195 15 × 14 = 210 15 × 15 = 225
16 × 1 = 16 16 × 2 = 32 16 × 3 = 48 16 × 4 = 64 16 × 5 = 80 16 × 6 = 96 16 × 7 = 112 16 × 8 = 128 16 × 9 = 144 16 × 10 = 160 16 × 11 = 176 16 × 12 = 192 16 × 13 = 208 16 × 14 = 224 16 × 15 = 240 16 × 16 = 256	17 × 1 = 17 17 × 2 = 34 17 × 3 = 51 17 × 4 = 68 17 × 5 = 85 17 × 6 = 102 17 × 7 = 119 17 × 8 = 136 17 × 9 = 153 17 × 10 = 170 17 × 11 = 187 17 × 12 = 204 17 × 13 = 221 17 × 14 = 238 17 × 15 = 255 17 × 16 = 272 17 × 17 = 289	18 × 1 = 18 18 × 2 = 36 18 × 3 = 54 18 × 4 = 72 18 × 5 = 90 18 × 6 = 108 18 × 7 = 126 18 × 8 = 144 18 × 9 = 162 18 × 10 = 180 18 × 11 = 198 18 × 12 = 216 18 × 13 = 234 18 × 14 = 252 18 × 15 = 270 18 × 16 = 288 18 × 17 = 306 18 × 18 = 324	19 × 1 = 19 19 × 2 = 38 19 × 3 = 57 19 × 4 = 76 19 × 5 = 95 19 × 6 = 114 19 × 7 = 133 19 × 8 = 152 19 × 9 = 171 19 × 10 = 190 19 × 11 = 209 19 × 12 = 228 19 × 13 = 247 19 × 14 = 266 19 × 15 = 285 19 × 16 = 304 19 × 17 = 323 19 × 18 = 342 19 × 19 = 361	20 × 1 = 20 20 × 2 = 40 20 × 3 = 60 20 × 4 = 80 20 × 5 = 100 20 × 6 = 120 20 × 7 = 140 20 × 8 = 160 20 × 9 = 180 20 × 10 = 200 20 × 11 = 220 20 × 12 = 240 20 × 13 = 260 20 × 14 = 280 20 × 15 = 300 20 × 16 = 320 20 × 17 = 340 20 × 18 = 360 20 × 19 = 380 20 × 20 = 400