Skip to main content
Home
plus.maths.org

Secondary menu

  • My list
  • About Plus
  • Sponsors
  • Subscribe
  • Contact Us
  • Log in
  • Main navigation

  • Home
  • Articles
  • Collections
  • Podcasts
  • Maths in a minute
  • Puzzles
  • Videos
  • Topics and tags
  • For

    • cat icon
      Curiosity
    • newspaper icon
      Media
    • graduation icon
      Education
    • briefcase icon
      Policy

      Popular topics and tags

      Shapes

      • Geometry
      • Vectors and matrices
      • Topology
      • Networks and graph theory
      • Fractals

      Numbers

      • Number theory
      • Arithmetic
      • Prime numbers
      • Fermat's last theorem
      • Cryptography

      Computing and information

      • Quantum computing
      • Complexity
      • Information theory
      • Artificial intelligence and machine learning
      • Algorithm

      Data and probability

      • Statistics
      • Probability and uncertainty
      • Randomness

      Abstract structures

      • Symmetry
      • Algebra and group theory
      • Vectors and matrices

      Physics

      • Fluid dynamics
      • Quantum physics
      • General relativity, gravity and black holes
      • Entropy and thermodynamics
      • String theory and quantum gravity

      Arts, humanities and sport

      • History and philosophy of mathematics
      • Art and Music
      • Language
      • Sport

      Logic, proof and strategy

      • Logic
      • Proof
      • Game theory

      Calculus and analysis

      • Differential equations
      • Calculus

      Towards applications

      • Mathematical modelling
      • Dynamical systems and Chaos

      Applications

      • Medicine and health
      • Epidemiology
      • Biology
      • Economics and finance
      • Engineering and architecture
      • Weather forecasting
      • Climate change

      Understanding of mathematics

      • Public understanding of mathematics
      • Education

      Get your maths quickly

      • Maths in a minute

      Main menu

    • Home
    • Articles
    • Collections
    • Podcasts
    • Maths in a minute
    • Puzzles
    • Videos
    • Topics and tags
    • Audiences

      • cat icon
        Curiosity
      • newspaper icon
        Media
      • graduation icon
        Education
      • briefcase icon
        Policy

      Secondary menu

    • My list
    • About Plus
    • Sponsors
    • Subscribe
    • Contact Us
    • Log in
    • Solution to Puzzle No. 3 - birth dates

      1 January, 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?

      • Log in or register to post comments
      University of Cambridge logo

      Plus Magazine is part of the family of activities in the Millennium Mathematics Project.
      Copyright © 1997 - 2025. University of Cambridge. All rights reserved.

      Terms