Monday Math 65

Divisibility Tests Part 5: Tests for 13

Divisibility tests for 13 are also quite rare, however we can create a test analogous to our first for 7.

We note that 9*10+1=91=7*13, so with n=10a+b and q=a-9b, we see
,
so 9n+q is a multiple of 13, which means that n is divisible by 13 if and only if q is divisible by 13. This gives us the test: mulitply the ones digit of our number by 9, and subtract from the number formed by the rest of the digits. We see that this doesn’t work well for two-digit numbers, but a little better for three- or four-digit numbers.
For example:
n=78
7-8*9=-65=-5*13
n=182
18-2*9=0=0*13
n=507
50-7*9=-13
n=3679
367-9*9=286
28-9*6=-26=-2*13
so all of these numbers are divisible by 13.

As with 7, this procedure may be laborious for very large numbers. However, we can see that
1001=7*143=7*11*13=77*13
This means 1000=13*77-1, and so we see our large-number divisibility test for 7 also works for 13.
Thus, we split our large number into sets of three digits, and then take the alternating sum of these numbers. For example:
n=348,970,129,352
352-129+970-348=845
84-5*9=39=3*13
so 348,970,129,352 is divisible by 13
n=4,626,145,317,014,712
712-14+317-145+626-4=1492
149-2*9=131=13*10+1, which is not divisible by 13,
so 4,626,145,317,014,712 is not divisible by 13.

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