a Z1 Come up with a criterion involving the digits of a for

a Z1. Come up with a criterion involving the digits of a for when a is divisible by 5 and a criterion for when a is divisible by 11. Prove your criteria!

Solution

Let a be any integer belonging to Z, the set of integers and a >=1

So a will be of the form 2,3,4,5......

Divisibility by 5:

If unit digit of a is ending in 5 or 0 then a is divisible by 5.

Since multiples of 5 will end only in 5 if multiplied with odd and 0 if multiplied with even, there is no other possibility for multiples of 5.

Hence a is divisible by 5 if unit digit = 5 or 0

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Divisibility by 11:

We know 11 divides 10+1, 100-1, 1000+1, 10000-1,.....

Thus alternate digits we add and find difference, If the difference is divisible by 11, the a is divisible by 11.

Example: 121

Alternate digits are 1+1 and 2

2-(1+1) = 0 divisible by 11 so 121 also

Similarly 12378

1+3+8 = 12 and 2+7 =9

Difference = 3 not divisible by 11 so 12378 is also not.