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King's Maths School

Challenge 409: A property of powers

Can you prove my conjecture?

Last week I was engaging in my hobby of typing numbers into my calculator at random when I noticed the following:

29 + 89 is divisible by 5

57 + 197 is divisible by 12

311 + 1111 is divisible by 7

63 + 203 is divisible by 13

What conjecture do you think I came up with?

Can you prove it? (And can you prove a slightly stronger conjecture?)

What happens with even powers? 

 

Submit your solution

Please do send in your solution to this problem to weeklymaths@kcl.ac.uk You can scan or photograph your written work, or type your solutions. If this is your first weekly maths challenge solution, please include your year group and the name of the school you attend. We'll be happy to provide feedback on your solution, assuming that you are in year 11 or below. If you are older than this, we hope you enjoy trying the problems and reviewing your solutions against those we publish on the website.