Solution to Puzzle No. 3 - birth dates

Submitted by plusadmin on January 1, 1998
January 1998

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,... , xn then

N = x[1]*10^(n-1) + x[2]*10^(n-2) + ..... x[n-1]*10 + x[n]

But it is always true that

10^m = 99...9 + 1

So we can write:

N = (x[1]*99...9 + x[2]*99..9 + ... + x[n-1]*9) + (x[1] + x[2] + ... x[n-1] + x[n])
N = 9*(x[1]*11...1 + x[2]*11..1 + ... + x[n-1]*1) + S

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?