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# Is the number 75 divisible by 2 3 4 5 9 10?

To be divisible by 2 the last digit must be even, ie one of {0, 2, 4, 6, 8};

The last digit of 75 is 5, which is not one of these so it is not divisible by 2.

To be divisible by 3, sum the digits of the number and if this sum is divisible by 3, then the original number is divisible by 3.

As the test can be repeated on the sum, repeat the summing until a single digit remains; only if this number is one of {3, 6, 9} is the original number divisible by 3.

For this gives:

75→7 + 5 = 12

12→1 + 2 = 3

3 is one of {3, 6,9} so it is divisible by 3.

To be divisible by 4, add the last (ones) digit to twice the previous (tens) digit; if this sum is divisible by 4, then so is the original number.

As the test can be repeated on the sum, repeat the summing until a single digit remains; only if this number is one of {4, 8} is the original number divisible by 4.

For this gives:

75→5 + 2×7 = 19

19→9 + 2×1 = 11

11→1 + 2×1 = 3

3 is not one of {4, 8} so it is not divisible by 4.

To be divisible by 5, the last digit must be one of {0, 5}.

The last digit of is 5 which is one of {0, 5} so it is divisible by 5.

To be divisible by 9, sum the digits of the number and if this sum is divisible by 9, then the original number is divisible by 9.

As the test can be repeated on the sum, repeat the summing until a single digit remains; only if this number is 9 is the original number divisible by 9.

For this gives:

75→7 + 5 = 12

12→1 + 2 = 3

3 is not 9 so it is not divisible by 9.

To be divisible by 10, the last digit must be 0

The last digit is 5 which is not 0, so it is not divisible by 10.

→ 75 is divisible by 3 and 5

75 is not divisible by 2, 4, 9, 10