Multiple Tricks
MULTIPLES OF 7
If we want to know whether a number larger than 70 is divisible by 7, the most obvious way is to divide by 7.
Another way is to work out a test number . If this number is a multiple of 7, then the original number will also be a multiple of 7.
Suppose that the number that you are investigating is N.
The units digit of N is U and the rest of N is R.
The test number, T, is 5U + R.
For example, if N = 98 then U = 8, R = 9, so T = 5 × 8 + 9 = 49 
a multiple of 7.
We can tackle larger values of N by repeating the process.
If N = 889 then U = 9, R = 88, so
T1 = 5 × 9 + 88 = 133, T2 = 5 × 3 + 13 = 28.
We can see why this is so by looking at N + 4T.
| N + 4T | = | (10R + U) + 4(5U + R) | = | 14R + 21U |
This is a multiple of 7, so if T is divisible by 7, so is N.
MULTIPLES OF 9
For a number N the test number is simply the sum of the digits of N. If N = 87264, then T = 8 + 7 + 2 + 6 + 4 = 27. Since T is divisible by 9, so is N . Why does this work?
If N has digits a, b, c, d, e, then T = a + b + c + d + e .
| N – T | = | (10 000a + 1000b + 100c + 10d) – (a + b + c + d) |
| = | 9999a + 999b + 99c + 9d |
MULTIPLES OF 11
To find the test number in this case alternately add and subtract the digits. For example, if N = 73645, then T = 7 – 3 + 6 – 4 + 5 = 11. < BR >
It follows that N is a multiple of 11.
If N has digits a, b, c, d, e, then T = a – b + c – d + e N – T = =
10 000a + 1000b +100c + 10d – a + b – c + d 9999a + 1001b + 99c + 11d
This is a multiple of 11, so both N and T are multiples of 11 or neither is.
MULTIPLES OF 13
For a number N the test for divisibility by 13 uses the test number T = 4U + R For instance, if N = 104 then U = 4, R = 10, so T = 4 x 4 + 10 = 26 divisible by 13.
To see why this works look at N + 3T. N + 3T =
(10R + U) + 3(4U + R) = 13R + 13U
This is multiple of 13, so if T is divisible by 13, so is N .
Pat Perkins
