SHORT CUTS

Divisibility Test (Quant)

 

Dividing by 2

1. All even numbers are divisible by 2. E.g., all numbers ending in 0,2,4,6 or 8.

Example:236,569874,369562

 

Dividing by 3

1.        Add up all the digits in the number.

2.        Find out what the sum is. If the sum is divisible by 3, so is the number

3.        For example: 12345 (1+2+3+4+5=15) 15  is divisible by 3, therefore 12345 is too!

Dividing by 4

1.        if the last two digits in your number divisible by 4. the number is too

2.        For example: 358912 ends in 12 which is divisible by 4, thus so is 358912.

Dividing by 5

1.        If the Numbers ending in a 5 or a 0 are always divisible by 5.

Dividing by 6

1.        If the Number is divisible by 2 and 3 it is divisible by 6 also.

Dividing by 7 (2 Tests)

• Take the last digit in a number.
• Double and subtract the last digit in your number from the rest of the digits.
• Repeat the process for larger numbers.
• Example: 357 (Double the 7 to get 14. Subtract 14 from 35 to get 21 which is divisible by 7 and we can now say that 357 is divisible by 7.

  NEXT TEST

• Take the number and multiply each digit beginning on the right hand side (ones) by 1, 3, 2, 6, 4, 5. Repeat this sequence as necessary
• Add the products.
• If the sum is divisible by 7 – so is your number.
• Example: Is 2016 divisible by 7?
• 6(1) + 1(3) + 0(2) + 2(6) = 21
• 21 is divisible by 7 and we can now say that 2016 is also divisible by 7.

Dividing by 8

1.        This one’s not as easy, if the last 3 digits are divisible by 8, so is the entire number.

2.        Example: 6008 – The last 3 digits are divisible by 8, therefore, so is 6008.

Dividing by 9

1.        Almost the same rule and dividing by 3. Add up all the digits in the number.

2.        Find out what the sum is. If the sum is divisible by 9, so is the number.

3.        For example: 43785 (4+3+7+8+5=27) 27 is divisible by 9, therefore 43785 is too!

Dividing by 10

1.        If the number ends in a 0, it is divisible by 10.

 

 

Squaring special numbers (7 and repeating 9’s)

Follow the following Steps:

1. Choose a number with a 7 and repeating 9’s (use a mimimum of three 9’s).
2. The square is made up of:

• first digits: 63 & two fewer 9’s than repeating 9’s in the number
• next digits: 84 & one fewer 0 than repeating 9’s in the number
• last digit: 1

 

Example-1:

1. If the number to be squared is 7999:
2. The square has:first digits: 63 and two fewer
9's than rep. 9's            6 3 9
next digits: 84 and one fewer
0 than rep. 9's                   8 4 0 0
last digit: 1                                 1
3. So 7999 × 7999 = 63,984,001.

Example-2:

1. If the number to be squared is 799999:
2. The square has:first digits: 63 and two fewer
9's than rep. 9's      6 3 9 9 9
next digits: 84 and one fewer
0 than rep. 9's                 8 4 0 0 0 0
last digit: 1                                  1
3. So 799999 × 799999 = 639,998,400,001.

Multiplying a repeating 2’s number by 66

 

Follow the following Steps:

1. Choose a repeating 2’s number (222, etc.)
2. Multiply it by 66
3. The product is:
   – the first two digits are 14
   – the next digits are repeating 6’s (two fewer than repeating 2’s in the original number)
   – the last two digits are 52

 

Example:1

1. If the first number is 222:
2. Multiply by 66.
3. The product is:
   – the first two digits are 14.
   – the next digits are repeating 6’s (two fewer than repeating 2’s in the original number): 6.
   – the last two digits are 52.
4. So 222 × 66 = 14652.

Example:2

1. If the first number is 22222:
2. The product is:
   – the first two digits are 14.
   – the next digits are repeating 6’s (two fewer than repeating 2’s in the original number): 666.
   – the last two digits are 52.
3. So 22222 × 66 = 1466652.

 

 

 

Multiplying a repeating 1’s number by 55

Follow the following Steps:

1. Choose a repeating 1’s number (111, etc.)
2. Multiply it by 55
3. The product is:
   – first digit is 6
   – next digits are repeating 1’s, two fewer than repeating 1’s in the original number
   – last two digits are 05.
4. If the first number is 1111:
5. Multiply by 55. The product is:
   first digit is 6: 6
   next digits are 1’s – two fewer than 1’s: 11
   last two digits are 05
6. So 1111 × 55 = 61105.

Example:1

Example:2

1. If the first number is 111111:
2. Multiply by 55. The product is:
   first digit is 6: 6
   next digits are 1’s, two fewer than 1’s: 1111
   last two digits are 05
3. So 111111 × 55 = 6111105.

 

Multiplying a repeating 1’s number by 25

Follow the following Steps:

1. Choose a repeating 1’s number (111, etc.)
2. Multiply it by 25
3. The product is:
   – first digit is 2
   – next digits are 7’s – one fewer than repeating 1’s in the original number
   – last digit is 5.

 

Example:1

1. If the first number is 11111:
2. Multiply by 25. The product is:
   first digit is 2: 2
   next digits are 7’s, one fewer than 1’s: 7777
   last digit is 5: 5
3. So 11111 × 25 = 277775.
4. If the first number is 11111111:
5. Multiply by 25. The product is:
   first digit is 2: 2
   next digits are 7’s, one fewer than 1’s: 7777777
   last digit is 5: 5
6. So 11111111 × 25 = 277777775

Example:2

 

Multiplying two 2-digit numbers (difference of 10)

 

Follow the following Steps:

1. Choose a 2-digit number.
2. Select a number either 10 smaller or 10 larger.
3. Find the middle number of the two (the average).
4. Square this middle number (multiply it by itself).
5. Subtract 25 from this square.

 

Example:1

1. If the first number is 36, choose 26 as the second number (10 smaller).
2. The middle number (the average) is 31.
3. Square this middle number: 31 × 31 = 961.
4. Subtract 25 from this square: 961 – 25 = 936
5. So 36 × 26 = 936.

Example:2

1. If the first number is 78, you might pick 88 as the second number (10 larger).
2. The middle number (the average) is 83.
3. Square this middle number: 83 × 83 = 6889.
4. Subtract 25 from this square: 6889 – 25 = 6864
5. So 78 × 88 = 6864.Remember to subtract in easy steps and pick your number to get an easy square.
   

Multiplying a 2-digit number by 125

 

Follow the following Steps:

1. Select a 2-digit number.
2. Divide the number by 8.
3. Move the decimal point 3 places to the right (add three zeros

 

Example:1

1. The 2-digit number chosen to multiply by 125 is 34.
2. Divide by 8: 34/8 = 4.25
3. Move the decimal point 3 places to the right: 4250
4. So 125 × 34 = 4,250.

Example:2

1. The 2-digit number chosen to multiply by 125 is 78.
2. Divide by 8: 78/8 = 9.75
3. Move the decimal point 3 places to the right: 9750
4. So 125 × 78 = 9,750.

Multiplying a repeating 9’s number by 88

 

Follow the following Steps:

1. Choose a repeating 9’s number (at least three digits long — 999, etc.)
2. Multiply it by 88
3. The product is:
   – the first two digits are 87
   – the next digits are repeating 9’s (two fewer than repeating 9’s in the original number)
   – the last two digits are 12
4. If the first number is 8888:
5. Multiply by 88. The product is:
   – the first two digits are 87: 87.
   – the next digits are repeating 9’s (two fewer than repeating 9’s in the original number): 99.
   – the last two digits are 12: 12.
6. So 9999 × 88 = 879912.

Example:

See the pattern?

1. If the first number is 999999:
2. Multiply by 88. The product is:
   – the first two digits are 87: 87.
   – the next digits are repeating 9’s (two fewer than repeating 9’s in the original number): 9999.
   – the last two digits are 12: 12.
3. So 999999 × 88 = 87999912.

 

Multiplying a 3-digit number by 101

Follow the Following Steps:

1. Select a 3-digit number .
2. The sum of the first and third digits will be the middle digit: _ _ X _ _.
3. The first two digits plus the carry will be the first digits: X X _ _ _.
4. The last two digits of the number will be the last digits: _ _ _ X X.
5. The number chosen is 318.
6. 3 + 8 = 11 (sum of first and third digits):
   _ _ 1 _ _ (keep carry, 1)
7. 31 + 1 = 32 (first two digits plus carry):
   3 2 _ _ _.
8. The last two digits are the same:
   _ _ _ 1 8.
9. So 318 × 101 = 32118.

Example:1

Example:2

1. If the number chosen is 728:
2. 7 + 8 = 15 (sum of first and third digits):
   _ _ 5 _ _ (keep carry, 1)
3. 72 + 1 = 73 (first two digits plus carry):
   7 3 _ _ _.
4. The last two digits are the same:
   _ _ _ 2 8.
5. So 728 × 101 = 73528.

 

Multiplying a 2-digit number by 101

 

Follow the following Steps:

1. Select a 2-digit number .
2. Write it twice!
   1. 47 × 101 = 4747.
   2. 38 × 101 = 3838.
   3. 96 × 101 = 9696.

Examples:

Multiplying a 2-digit number by 999

Follow the following Steps:

1. Select a 2-digit number.
2. Add one zero to the number, subtract 1:
   X X X _ _
3. Subtract the original number from 100:
   _ _ _ X X
4. The 2-digit number chosen to multiply by 999 is 64.
5. Add one zero and subtract 1: 640 – 1 = 639:
   6 3 9 _ _
6. Subtract 64 from 100: 100 – 64 = 36:
   X X X 3 6
7. So 64 × 999 = 63,936.

Example:1

Example:2

1. The 2-digit number chosen to multiply by 999 is 75.
2. Add one zero and subtract 1: 750 – 1 = 749:
   7 4 9 _ _
3. Subtract 75 from 100: 100 – 75 = 25:
   X X X 2 5
4. So 64 × 999 = 74,925.

 

Dividing a 2-digit number by 875

Follow the following Steps:

1. Select a 2-digit number.
2. Multiply it by 8.
3. Divide by 7.
4. Move the decimal point 3 places to the left.
5. The 2-digit number chosen to divide by 875
   is 31.
6. Multiply by 8: 31 × 8 = 248
7. Divide by 7: 248/7 = 35 3/7
8. Move the decimal point 3 places to the left: .035 3/7
9. So 31 divided by 875 = .035 3/7.
10. The 2-digit number chosen to divide by 875
    is 63.
11. Multiply by 8: 8 × 63 = 480 + 24 =
    480 + 20 + 4 = 500 + 4 = 504
12. Divide by 7: 504/7 = 72
13. Move the decimal point 3 places to the left: .072
14. So 63 divided by 875 = .072.

Finding the square root of 2-digit numbers ending in 1

 

Follow the following Steps:

1. Select a 2-digit number and square it.
2. Drop the last two digits of the square.
3. Find the largest square root of the remaining digits.
   This is first digit of the square root.
4. The second digit is 1.
5. If the square is 2601:
6. Drop the last two digits: 26
7. Find the largest root in 26: 5 × 5 = 25
8. The first digit is 5. The second digit is 1.
9. So the square root of 2601 is 51.
10. If the square is 8281:
11. Drop the last two digits: 82
12. Find the largest root in 82: 9 × 9 = 81
13. The first digit is 9. The second digit is 1.
14. So the square root of 2601 is 91.

Examples:1

Examples:2

Finding the square root of perfect squares ending in 5

 

Follow the following Steps:

1. Select a 2-digit number and square it.
2. Drop the last two digits of the square.
3. Find the largest square root of the remaining digits.
   This is first digit of the square root.
4. The second digit is 5.

Example:1

1. If the square is 9025:
2. Drop the last two digits: 90
3. Find the largest root in 90: 9 × 9 = 81
4. The first digit is 9. The second digit is 5.
5. So the square root of 9025 is 95.

Example:2

 

1. If the square is 4225:
2. Drop the last two digits: 42
3. Find the largest root in 42: 6 × 6 = 36
4. The first digit is 6. The second digit is 5.
5. So the square root of 4225 is 65

Dividing a 2-digit number by 625

 

Follow the Following Steps:

1. Select a 2-digit number.
2. Multiply by 8.
3. Divide the product by 5.
4. Move the decimal point 3 places to the left.
5. The 2-digit number chosen to divide by 625 is 65.
6. Multiply by 8: 8 × 65 = 480 + 40 = 520
7. Divide by 5: 520/5 = 104
8. Move the decimal point 3 places to the left: .104
9. So 65 divided by 625 = .104.

Example:1

Example:2

1. The 2-digit number chosen to divide by 625 is 32.
2. Multiply by 8: 8 × 32 = 240 + 16 = 256
3. Divide by 5: 256/5 = 51.2
4. Move the decimal point 3 places to the left: .0512
5. So 32 divided by 625 = .0512.

Multiplying a 2-digit number by 39

Follow the Following Steps:

1. Select a 2-digit number.
2. Multiply by 4.
3. Add one zero.
4. Subtract the original number.
5. The 2-digit number chosen to multiply by 39 is 21.
6. Multiply by 4: 4 × 21 = 84.
7. Add one zero: 840.
8. Subtract the original number:
   840 – 21 = 819.
9. So 21 × 39 = 819.
10. The 2-digit number chosen to multiply by 39 is 71.
11. Multiply by 4: 4 × 71 = 284.
12. Add one zero: 2840.
13. Subtract the original number:
    2840 – 71 = 2800 – 31 = 2769.
14. So 71 × 39 = 2769.

Example:1

Example:2

 

 

Multiplying a 2-digit number by 29

Follow the Following Steps:

1. Multiply by 3.
2. Add one zero.
3. From this, subtract the original number.

Example:1

1. The 2-digit number chosen to multiply by 29 is 21.
2. Multiply by 3: 3 × 21 = 63
3. Add one zero: 630
4. Subtract original number:
   630 – 21 = 610 – 1 = 609
5. So 21 × 29 = 609.

Example:2

1. The 2-digit number chosen to multiply by 29 is 81.
2. Multiply by 3: 3 × 81 = 243
3. Add one zero: 2430
4. Subtract the original number:
   2430 – 81 = 2350 – 1 = 2349
5. So 81 × 29 = 2349.

 

Finding 55 percent of a number

Follow the following Steps:

1. Choose a 2-digit number.
2. Multiply the number by 11.
3. Divide by 2.
4. Move the decimal point one place to the left.
5. If the number selected is 81:
6. Multiply by 11: 11 × 81 = 891
   right digit is 1
   next digit to left is 1 + 8 = 9
   last digit to left is 8
7. Divide by 2: 891/2 = 445.5
8. Move the decimal point one place to the left: 44.55
9. So 55% of 81 = 44.55.

Example-1:

Example-2:

1. If the number selected is 59:
2. Multiply by 11: 11 × 59 = 649
   right digit is 9
   next digit to left is 9 + 5 = 14 (use the 4 and carry 1)
   last digit to left is 5 + 1 = 6
3. Divide by 2: 649/2 = 324.5
4. Move the decimal point one place to the left: 32.45
5. So 55% of 59 = 32.45.

Finding 45 percent of a number

Follow the following Steps:

1. Choose a 2-digit number.
2. Multiply the number by 9.
3. Divide by 2.
4. Move the decimal point one place to the left.
5. If the number selected is 36:
6. Multiply by 9: 9 × 36 = 270 + 54 = 324
7. Divide by 2: 324/2 = 162
8. Move the decimal point one place to the left: 16.2
9. So 45% of 36 = 16.2.
10. If the number selected is 52:
11. Multiply by 9: 9 × 52 = 450 + 18 = 468
12. Divide by 2: 468/2 = 234
13. Move the decimal point one place to the left: 23.4
14. So 45% of 52 = 23.4.

Example-1:

Example-2:

Finding 40 percent of a number

Follow the Following Steps:

1. Choose a 2-digit number.
2. Multiply the number by 4.
3. Move the decimal point one place to the left.

Example-1:

1. If the number selected is 21:
2. Multiply by 4: 4 × 21 = 84
3. Move the decimal point one place to the left: 8.4.
4. So 40% of 21 = 8.4.

Example-2:

1. If the number selected is 73:
2. Multiply by 4: 4 × 73 = 280 + 12 = 292.
3. Move the decimal point one place to the left: 29.2.
4. So 40% of 73 = 29.2.

 

 

 

 

(88)