Solution to Puzzle No. 3 - birth dates

Share this page
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?

  • Want facts and want them fast? Our Maths in a minute series explores key mathematical concepts in just a few words.

  • What do chocolate and mayonnaise have in common? It's maths! Find out how in this podcast featuring engineer Valerie Pinfield.

  • Is it possible to write unique music with the limited quantity of notes and chords available? We ask musician Oli Freke!

  • How can maths help to understand the Southern Ocean, a vital component of the Earth's climate system?

  • Was the mathematical modelling projecting the course of the pandemic too pessimistic, or were the projections justified? Matt Keeling tells our colleagues from SBIDER about the COVID models that fed into public policy.

  • PhD student Daniel Kreuter tells us about his work on the BloodCounts! project, which uses maths to make optimal use of the billions of blood tests performed every year around the globe.