Lesson11

Lesson11

Can it Divide?

How can you quickly know if one number will divide evenly into another number, leaving no remainder? For example, will 3
divide evenly into 2,169,252? Well, I wouldn't have brought the subject up if I didn't know some curious shortcuts.I think all
of you know the division by 2 ,5 ,10 . Now we learn division by every possible number
Division by 2
No surprise here. Any number that ends in 0,2,4,6 or 8 is evenly divisible by 2.
Division by 3
Add the number's digits. If the sum is evenly divisible by 3, then so is the number. So, will 3 divide evenly into 2,169,252?
Yes it will, because the sum of the digits is 27, and 27 is divisble by 3. If you want, you can keep adding numbers until one
digit remains. For example, keep going with 27. 2 + 7 = 9, which is also evenly divisible by 3.
Division by 4
If the number's last 2 digits are 00 or if they form a 2-digit number evenly divisible by 4, then number itself is divisible by 4.
How about 56,789,000,000? Last 2 digits are 00, so it's divisible by 4. Try 786,565,544. Last 2 digits, 44, are divisible by 4
so, yes, the whole number is divisible by 4.
Division by 5
Any number that ends in a 0 or 5 is evenly divisible by 5. Easy enough.
Division by 6
The number has to be even. If it's not, forget it. Otherwise, add up the digits and see if the sum is evenly divisible by 3. It it
is, the number is evenly divisible by 6. Try 108,273,288. The digits sum to 39 which divides evenly into 13 by 3, so the
number is evenly divisible by 6. If you want, you can keep adding numbers until only one digit remains and do the same
thing. So in this case, 3 + 9 = 12 and 1 + 2 = 3, and 3 is evenly divisible by 3!
Division by 7
Multiply the last digit by 2. Subtract this answer from the remaining digits. Is this number evenly divisible by 7? If it is, then
your original number is evenly divisible by 7. Try 364. 4, the last digit, multiplied by 2 = 8. 36, the remaining digits,
minus 8 = 28. The last time I checked, 28 is evenly divisble by 7, and thus, so is 364!
Another example 1792 2 is the last digit multiply it by 2and subtract it from original number
1792
-4
1 7 5 repeat once again 5x2=10
- 10
7 7 is divisible by 7 so is 1792
Division by 8
If the number's last 3 digits are 000 or if they form a 3-digit number evenly divisible by 8, then the number itself is divisible
by 8. How about 56,789,000,000? Last 3 digits are 000, so it's divisible by 8. Try 786,565,120. The last 3 digits, 120, divide
by 8 into 15, so yes, the whole number is divisible by 8.
Division by 9
Sum the number's digits. If it divides by 9, you're in luck. As with the tests for 3 and 6, you can keep adding numbers until
you're left with only one digit. 9873 , 9+8+7+3=27 =2+7 =9 9873 is divisible by 9
Division by 10
Any number that ends in 0 is evenly divisible by 10.
Division by 11
Here are four ways for different types of numbers:
1. If the sum of every other digit, starting with the first, is equal to the sum of every other digit starting with the
second, then the number is evenly divisible by 11. Try 13057. 1+0+7 = 3+5, therefore it should divide evenly
by 11. And indeed it does: 13057 / 11 = 1187. 1+0+7-3-5=0
2. If the digits are different, count them from the right and then add the numbers in the odd positions and the
even positions. Subtract the smaller number from the larger. If the difference is evenly divisible by 11, so is
your original number. Take the number 181,907. The numbers 8,9, and 7 are in the odd positions. They sum
to 24. The numbers 1,1, and 0 are in the even positions. They sum to 2. Subtract 2 from 24 to get 22. 22
divides by 11 into 2, so 181,907 is evenly divisible by 11.
Division by 12
If the number can be evenly divided by 3 and 4, the same can also be said for 12. Use the methods for Division by 3 and
Division by 4 above. If they both work, your number is also evenly divisible by 12.
Division by 13
Multiply the last digit by 4 and add it from remaining digits Is this number evenly divisible by 13? If it is, then your original
number is evenly divisible by 13. Try 598
598
+32 (8x4=32)
9 1 Repeat once again
+ 4
13 so it is divisible by 13
Division by 15
If the number can be evenly divided by 3 and 5, the same can also be said for 15. Use the methods for Division by 3 and
Division by 5 above. If they both work, your number is also evenly divisible by 15.
Division by 17
Multiply the last digit by 5. Subtract this answer from the remaining digits. Is this number evenly divisible by 17? If it is, then
your original number is evenly divisible by 17. Try 663
663
- 15
51 it is divisible by 17 so 663 is divisible by 17
Division by 19
Multiply the last digit by 2. Add this answer from the remaining digits. Is this number evenly divisible by 19? If it is, then
your original number is evenly divisible by 19. Try 741
741
+ 2
76
+ 12
19 it is divisible by 19 so 741 is divisible by 19
Division by 23
Multiply the last digit by 7. Add this answer from the remaining digits. Is this number evenly divisible by 23? If it is, then
your original number is evenly divisible by 23. Try 667
667
+ 47
113

+ 21
23 it is divisible by 23 so 667 is divisible by 23
Division by 24
If the number can be evenly divided by 3 and 8, the same can also be said for 24. Use the methods for Division by 3 and
Division by 8 above. If they both work, your number is also evenly divisible by 24.
Division by 29
Multiply the last digit by 3. Add this answer from the remaining digits. Is this number evenly divisible by 29? If it is, then
your original number is evenly divisible by 19. Try 667
667
+ 21
87
+ 21
29 it is divisible by 29 so 667 is divisible by 29

Division by 31
Multiply the last digit by 3. Subtract this answer from the remaining digits. Is this number evenly divisible by 31? If it is, then
your original number is evenly divisible by 31. Try 744
744
- 1 2
62
6
0 it is divisible by 31 so 744 is divisible by 31
Division by 33
If the number can be evenly divided by 3 and 11, the same can also be said for 33. Use the methods for Division by 3 and
Division by 11 above. If they both work, your number is also evenly divisible by 33.
Division by 36
If the number can be evenly divided by 4 and 9, the same can also be said for 36. Use the methods for Division by 4 and
Division by 9 above. If they both work, your number is also evenly divisible by 36.
Division by 37
Multiply the last digit by 11. Subtract this answer from the remaining digits. Is this number evenly divisible by 37? If it is,
then your original number is evenly divisible by 37. Try 925
925
- 55
37 it is divisible by 31 so 925 is divisible by 37

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Lesson10

Lesson10
Squares

Finding square of any number is very very easy and you can calculate it very fast But before
going to concept ,you should do some home work . You are well conversant in squaring of
1 to 9
1x1 =1
2x2 =4
3x3 =9
4x4 =16
5x5 =25
6x6 =36
7x7 =47
8x8 =64
9x9 =81
These are square from 1 to 9 . Now learn square index of 1 to 9
Number Square Index Operation
1 2 First digit
squaring
1 to4
Multiply first
digit to next to
first digit
5 to9
2 4
3 6
4 8
5 0
6 2
7 4
8 6
9 8
Square Index is nothing except table of 2 .We just double
number in first part ie from 1 to 4 and leave ten 's digit in
second part ie from 5 to 9 after doubling .
Steps Now learn main concept by this example
21 x21 =????
Here our last digit (or unit digit ) is1 .Its square index
is 2 First we take square of 1 and write it extreme right then
we take first digit and multiply it by the square index write before
last figure then squaring first digit and write extreme left
If there is any carry add to it
1x1 =1 2x2 =4 22 =4 4 4 1 Answer
44 X44=????
Here square index of 4 is 8
4x4 = 16 4x8 +1(carry ) =32 +1 = 33 42 + 3 =16+3 =19 1936 Answer
93 X93 =????
Here square index of 3 is 6
3x3 =9 9x6 =54 92+ 5 =81 +5 =86 8649 Answer
These are the square where last digit is 1 to 4 now we learn squaring of those numbers
having last digit from 5 to 9.Method is same, instead of squaring first digit multiply to
next digit
36 X36 =
Here square index of 6 is 2
6x6 =36 3x2 + 3 =6+3=9 6X(6+1) = 6 x7 =42 42 96 answer
87 x 87 =
Here square index is 4
7x7=49 8x4+4 =32+4 =36 8x(8+1)+3 =72+3 =75 75 69 Answer
25x25 =
Here square index is 0 so no need of middle step
5x5 = 25 2x(2+1) = 6 625 answer
Now with the help of this concept we can calculate square of any number
236 x 236=????
Here square index of 6is 2
6x6 = 36 23 x2 +3 =46+3=49 23x24*+4 =552+4=556 55696 answer
*23x24 = ? By cross and vertical multiplication method(covered in lesson no 7)
4 / 14 /12 = 552

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