Maths Problem

[saladdodger]

I am trying to find square numbers which divide by 3, 5 and 7. Is there a quick method please?

[jim360]

In fact if a square number is divisible by 3, then it must also be divisible by 3^2 = 9, and so on. So you are looking for numbers that are divisible not only by 3, 5, 7 but also 3 twice, 5 twice and 7 twice.

If it's these numbers individually then you can take 3^2 = 9, (3^2)*(2^2) = 9*4 = 36 etc. along with 5^2= 25, (5^2*)*(2^2) = 25*4 = 100 etc.

Otherwise you are looking for square numbers which are multiples of (3^2)*(5^2)*(7^2) = 9*25*49 = 11025, and then take square multiples of this.

The key point then is that square numbers have each distinct prime number as a factor an even number of times (0,2,4,6 times... but always an even number of the same prime factor).

[saladdodger]

Ah yes thanks for that - I'll look at multiples of 11025 for my required answer then

[jim360]

What's the context of the problem by the way?

[saladdodger]

It's from prospect generalist quizzes and puzzles - I need a 6 didgit square number that will divide by 3, 5 and 7 and all its digits have to be different. There would seem to be more than one answer.

[jim360]

Ah OK. I checked and there's only the one answer.

[saladdodger]

yes i've also had a rethink and found there's only 1 answer. Thanks for your help with the 3x3 5x5 and 7x7 bit earlier.