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14 Divisors

What is the smallest number with exactly 14 divisors?

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Repeaters

Choose any 3 digits and make a 6 digit number by repeating the 3 digits in the same order (e.g. 594594). Explain why whatever digits you choose the number will always be divisible by 7, 11 and 13.

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Oh! Hidden Inside?

Find the number which has 8 divisors, such that the product of the divisors is 331776.

Differences

Stage: 3 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

There were many solutions to the first part of this question based on odd and even numbers.

Alexander from Shevah-Mofet School, Israel and also the Key Stage 3 Maths Club from Strabane Grammar School proved that, for any distinct whole numbers a , b and c , the expression

( a - b )( a - c )( b - c ) is divisible by 2.

At least two of the distinct whole numbers a , b or c must have the same remainder when divided by 2 (because there are only two possibilities), therefore their difference will be an even number and ( a - b )( a - c )( b - c ) must be even

Alexander generalised this solution:

With four distinct whole numbers the same idea is used to prove that( a - b )( a - c )( a - d )( b - c )( b - d )( c - d) is divisible by 3. At least two of the numbers a , b , c or d have the same remainder when divided by 3; therefore one of the differences divides by 3 and so the entire thing divides by 3.

Taking n distinct whole numbers, the product of the differences of the numbers taken in pairs is divisible by ( n -1) because at least two of the numbers have the same remainder when divided by ( n -1).

The Maths Club at Wilson's School sent in this very clearly explained proof:

If we have a set of n+1 integers, then by the pigeonhole principle, two of these integers must have the same value modulo n. Therefore the difference between these 2 values will be congruent to 0 modulo n, and so will be a multiple of n. Thus the product of all the remainders will be a multiple of n.

Also because in a set of n+1 integers you always have 2 values that the same modulo n-1, n-2, n-3 ... , 2 , 1, we can say that the product of the differences of n+1 numbers will be divisible by n!

Ruth from The Manchester High School for Girls proved that the product of the differences of four numbers is not necessarily a multiple of 5. For example the 4 numbers 1, 2, 3 and 4. The product of the differences is (2-1)(3-2)(3-1)(4-3)(4-2)(4-1)=1*1*2*1*2*3=12 which is not a multiple of 5.

For further discussion of this problem see Modulus Arithmetic and a Solution to Differences by Peter Zimmerman (Mill Hill County High School, London). Peter has given a very good account of modulus arithmetic which not only provides a solution to this problem but will also help you to understand a method which can be applied to many other problems.