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56 = 7 × 8, thus the number must be divisible by both 7 and 8 - check for each of these digits:

8 is easy to check: add the units digit to twice the tens digit to 4 times the hundreds digit; if this sum is divisible by 8 then so is the original number.

There is no easy check for 7. One I can offer:

Split the number into blocks of 3 digits starting at the right hand end.

Now alternatively subtract and add the each of the digits from the blocks: the units, tens and hundreds digits separately.

Finally add the sum of the units digits to 3 times the sum of the tens digits to twice the sum of the hundreds digits; if this sum is divisible by 7, then so is the original number.

eg 123456789 → 123 456 789

→ sum units: 9 - 6 + 3 = 6

→ sum tens: 8 - 5 + 2 = 5

→ sum hundreds: 7 - 4 + 1 = 4

→ check sum: 6 + 3×5 + 2×4 = 29

29 is not divisible by 7, so 123456789 is not divisible by 7.

[The remainder of 29 divided by 7 is 1, so 123456789 divided by 7 has a remainder of 1.]

eg is 135792648 divisible by 56?

8 + 2×4+4×6 = 40 which is divisible by 8, so 135792648 is divisible by 8.

135792648 → 135 792 648

→ sum units: 8 - 2 + 5 = 11

→ sum 10s: 4 - 9 + 3 = -2

→ sum 100s: 6 - 7 + 1 = 0

→ chk: 11 + 3×-2 + 2×0 = 5 not divisible by 7, so 135792648 is not divisible by 7

Thus 135792648 is not divisible by 56

eg is 63592648 divisible by 56?

Divisible by 8 as before: 8 + 2×4 + 4×6 = 40 = 5×8

63 592 648

→ sum units: 8 - 2 + 3 = 9

→ sum 10s: 4 - 9 + 6 = 1

→ sum 100s: 6 - 5 = 1

→ chk: 9 + 3×1 + 2×1 = 14 = 2×7 → divisible by 7

→ 63592648 is divisible by 56

(63592648 = 113558×56)

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Q: What are the divisibility rules for 56?
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How is understanding factors help you write divisibility rules?

It's not completely necessary to know the factors if the number ends in 1, 3, 7 or 9. You can sum or subtract a certain number of times the last digit by the rest of the number if the number ends in 1, 3, 7 or 9. However I think it's required to factorize the number if it ends in 0, 2, 4, 5, 6 or 8. Here are the divisibility rules of every number from 1 to 50 1: Every number is a multiple of 1 2: The number ends in 0, 2, 4, 6 or 8 3: The sum of the digits is a multiple of 3 4: The last 2 digits are a multiple of 4 The 10s digit is even and the last digit is 0, 4 or 8 The 10s digit is odd and the last digit is 2 or 6 5: The number ends in 0 or 5 6: The number is a multiple of 2 and 3 at the same time 7: The difference between twice the last digit and the rest of the number is a multiple of 7 8: The last 3 digits are a multiple of 8 The 100s digit is even and the last 2 digits are a multiple of 8 The 100s digit is odd and the last 2 digits are 4 times an odd number 9: The sum of the digits is a multiple of 9 10: The number ends in 0 11: The difference between the last digit and the rest of the number is a multiple of 11 12: The number is a multiple of 3 and 4 at the same time 13: The sum of 4 times the last digit and the rest of the number is a multiple of 13 14: The number is a multiple of 2 and 7 at the same time 15: The number is a multiple of 3 and 5 at the same time 16: The last 4 digits are a multiple of 16 The 1,000s digit is even and the last 3 digits are a multiple of 16 The 1,000s digit is odd and the last 3 digits are 8 times an odd number 17: The difference between 5 times the last digit and the rest of the number is a multiple of 17 18: The number is a multiple of 2 and 9 at the same time 19: The sum of twice the last digit and the rest of the number is a multiple of 19 20: The number ends in 00, 20, 40, 60 or 80 21: The difference between twice the last digit and the rest of the number is a multiple of 21 22: The number is a multiple of 2 and 11 at the same time 23: The sum of 7 times the last digit and the rest of the number is a multiple of 23 24: The number is a multiple of 3 and 8 at the same time 25: The number ends in 00, 25, 50 or 75 26: The number is a multiple of 2 and 13 at the same time 27: The difference between 8 times the last digit and the rest of the number is a multiple of 27 28: The number is a multiple of 4 and 7 at the same time 29: The sum of thrice the last digit and the rest of the number is a multiple of 29 30: The number is a multiple of 3 and 10 at the same time 31: The difference between thrice the last digit and the rest of the number is a multiple of 31 32: The last 5 digits are a multiple of 32 The 10,000s digit is even and the last 4 digits are a multiple of 32 The 10,000s digit is odd and the last 4 digits are 16 times an odd number 33: The sum of 10 times the last digit and the rest of the number is a multiple of 33 34: The number is a multiple of 2 and 17 at the same time 35: The number is a multiple of 5 and 7 at the same time 36: The number is a multiple of 4 and 9 at the same time 37: The difference between 11 times the last digit and the rest of the number is a multiple of 37 38: The number is a multiple of 2 and 19 at the same time 39: The sum of 4 times the last digit and the rest of the number is a multiple of 39 40: The last 3 digits are a multiple of 40 The 100s digit is even and the last 2 digits are 00, 40 or 80 The 100s digit is odd and the last 2 digits are 20 or 60 41: The difference between 4 times the last digit and the rest of the number is a multiple of 41 42: The number is a multiple of 2 and 21 at the same time 43: The sum of 13 times the last digit and the rest of the number is a multiple of 43 44: The number is a multiple of 4 and 11 at the same time 45: The number is a multiple of 5 and 9 at the same time 46: The number is a multiple of 2 and 23 at the same time 47: The difference between 14 times the last digit and the rest of the number is a multiple of 47 48: The number is a multiple of 3 and 16 at the same time 49: The sum of 5 times the last digit and the rest of the number is a multiple of 49 50: The number ends in 00 or 50


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