Skip over navigation
Cambridge University Faculty of Mathematics NRich logo
menu search
  • Teachers expand_more
    • Early years
    • Primary
    • Secondary
    • Post-16
    • Events
    • Professional development
  • Students expand_more
    • Primary
    • Secondary
    • Post-16
  • Parents expand_more
    • Early Years
    • Primary
    • Secondary
    • Post-16
  • Problem-Solving Schools
  • About NRICH expand_more
    • About us
    • Impact stories
    • Support us
    • Our funders
    • Contact us
  • search

Or search by topic

Number and algebra

  • The Number System and Place Value
  • Calculations and Numerical Methods
  • Fractions, Decimals, Percentages, Ratio and Proportion
  • Properties of Numbers
  • Patterns, Sequences and Structure
  • Algebraic expressions, equations and formulae
  • Coordinates, Functions and Graphs

Geometry and measure

  • Angles, Polygons, and Geometrical Proof
  • 3D Geometry, Shape and Space
  • Measuring and calculating with units
  • Transformations and constructions
  • Pythagoras and Trigonometry
  • Vectors and Matrices

Probability and statistics

  • Handling, Processing and Representing Data
  • Probability

Working mathematically

  • Thinking mathematically
  • Developing positive attitudes
  • Cross-curricular contexts

Advanced mathematics

  • Decision Mathematics and Combinatorics
  • Advanced Probability and Statistics
  • Mechanics
  • Calculus

For younger learners

  • Early Years Foundation Stage

Inclusion Exclusion

Age 11 to 14
Challenge Level Yellow starYellow starYellow star
  • Problem
  • Getting Started
  • Student Solutions
  • Teachers' Resources

How many integers between 1 and 1200 are NOT multiples of any of the numbers 2, 3 or 5?

Venn diagram showing whole numbers from 1 to 1200

Working along the same lines, Emma and Laura from The Mount School in York sent the diagram above, and Lorn from Stamford School sent the solution below:

Simplify the problem by trying 1-30 (because 2 x 3 x 5 = 30 and 30 is a factor of 1200).

First I crossed off all the multiples of 2 (even numbers). Then I crossed off all multiples of 3, then all multiples of 5. I then counted 8 numbers which were not crossed off.

I tried 31-60 and found a pattern.
Comparing 31-60 and 1-30, the crossed off numbers corresponded.
This is not surprising since you get the numbers in the second set just by adding 30 to the numbers in the first set.
I also noticed that the numbers I did not cross off were prime numbers and 2, 3 and 5 are prime.

There are 40 sets of 30 numbers from 1 to 1200 (1-30, 31-60, 61-90 ... ).
From this I conclude that in each set there are 8 numbers that are not multiples of 2, 3 or 5, so there are 8 x 40 = 320 numbers altogether that are not multiples of 2, 3 or 5.

Lorn found another method, using similar ideas, sparked from the Venn diagram shown in the original question:

The diagram shows three sets of numbers: multiples of 2, multiples of 3 and multiples of 5.
When counting all the numbers in the three sets (600 + 400 + 240 = 1240) some numbers are counted twice and some numbers are counted 3 times.
Deducting the number of 'double counted' numbers we get 1240 - 200 - 80 - 120 = 840. Now the 'triple counted' numbers have been removed 3 times so we need to add on 40 to include them, giving 880 numbers which are multiples of one or more of the given numbers.
It follows that 1200 - 880 = 320 numbers are not multiples of any of these numbers.
 

Congratulations for your work on this problem to Samantha of Hethersett High School, Norfolk, Jenny, Caroline, Emma, Rachel and Beth from the Mount School, York and to Ben.

You may also like

Adding All Nine

Make a set of numbers that use all the digits from 1 to 9, once and once only. Add them up. The result is divisible by 9. Add each of the digits in the new number. What is their sum? Now try some other possibilities for yourself!

Have You Got It?

Can you explain the strategy for winning this game with any target?

Counting Factors

Is there an efficient way to work out how many factors a large number has?

  • Tech help
  • Accessibility Statement
  • Sign up to our newsletter
  • Twitter X logo

The NRICH Project aims to enrich the mathematical experiences of all learners. To support this aim, members of the NRICH team work in a wide range of capacities, including providing professional development for teachers wishing to embed rich mathematical tasks into everyday classroom practice.

NRICH is part of the family of activities in the Millennium Mathematics Project.

University of Cambridge logo NRICH logo