Is there a divisibility test for the 7 times table?

Answer 1

Yes, there is a divisibility test known as casting out 7s. It is based on congruences which forms part of Mathematical Number Theory. The process is complicated and is of limited practical use.

When you divide 10 by 7 there is a remainder of 3, when 100 is divided by 7 the remainder is 2.

This is written 10 Ξ 3, 10² Ξ 2, 10³ Ξ -1, 10⁴Ξ -3, 10⁵ Ξ -2, 10⁶ Ξ 1 (mod 7)

Consequently, N Ξ a₀ + 3a₁+ 2a₂-a₃-3a₄-2a₅ + a₆ .....(mod 7)

The process is best illustrated with an example : To check if 141589 is divisible by 7 then:-

N Ξ 141589 Ξ 9 + 3x8 +2x5 -1 -3x4 -2x1 = 9 + 24 + 10 - 1 - 12 - 2 = 28 Ξ 0 (mod 7)

The final part of the statement 28 Ξ 0 (mod 7) simply tell you that 28 is wholly divisible by 7 and there is no remainder. Consequently 141589 is divisible by 7.