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*=10*a*+*b* and *q*=*a*-9*b*, we see

,

so 9*n*+*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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Tags: Divisibility, Divisibility Tests, Math, Monday Math, Number Theory

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