Solution to Puzzle No. 3 - birth dates

For the question see "Puzzle No. 3 - birth dates", in issue 3.

The answer to the problem is that, in order to share this special numerical relationship, the age of the mother must be a multiple of 9 when the child is born.

Why?

To prove that this is the case, we need a little theorem.

Theorem

If N is a positive integer and S is the sum of its digits, then N mod 9 = S mod 9.

"N mod 9" just means "the remainder r when N is divided by 9", where r can range from 0 to 8.

Proof

The proof, in a nutshell, looks like this:

If the digits of N are x₁, x₂,... , x_n then

But it is always true that

So we can write:

But the first term of this expression is an exact multiple of 9.
Therefore N mod 9 = S mod 9.

QED.

This theorem tells us that summing the digits of a number does not change its value mod 9. Therefore, repeatedly summing the digits of a number until a single digit is reached does not change its value mod 9.

On the day a child is born its age is 0. Therefore, to share this special numerical relationship with its parent the parent's age mod 9 must also be 0. This is simply another way of saying that the parent's age must be a multiple of 9, e.g., 9, 18, 27, 36, 45, 54,...

Perhaps a more famous use of this theorem is in deducing that a number is divisible by 3 if and only if the sum of its digits is divisible by three. Why?