Divisible by all!

Number Theory Level 4

Find the sum of all positive integers $n$ such that for all odd integers $a$ , if $a^2\leq n$ then $n$ is divisible by $a$ .

I just slightly changed a problem from the book Number Theory: Structures, Examples and Problems which I think is quite interesting.

1 solution

Soumya Chakraborty
Feb 16, 2014

All the multiples of 1: from $1^{2}$ to ( $3^{2}$ - 1), viz. 1 to 8 ... All these satisfy

All the multiples of 3: from 9 to 24, viz. 9, 12, 15,... 24

All the multiples of LCM(3,5): from 25 to 48, viz. 30 and 45

In no other range will we get any other number.

So, sum of all the numbers identified = $\boxed{210}$

did it by logic

Bart Khau - 7 years, 3 months ago

I think there can be more numbers , for example 35, 40 for a=5 and 56, 63 for a=7 and many more. Would u please explain ?

Anil Chahal - 7 years, 2 months ago

It needs to hold true for all positive odd integers $a$ where $a^2 \leq n$ .

For $n=35$ , the odd integers $a$ are a=1, 3, 5, -1, -3, -5. In this case, 35 is not divisible by 3 (or -3)

Calvin Lin Staff - 7 years, 2 months ago

Thank You. I got it now.

Anil Chahal - 7 years, 2 months ago