Squares from non squares

Probability Level pending

You have three different positive integers, $a$ , $b$ , and $c$ .

None of them are square numbers.

However, if you multiply any two together you get a square number.

What is the smallest possible value for $a+b+c$ ?

The answer is 28.

3 solutions

Kushal Bose
Dec 2, 2016

We can get many such integers like this:

Let $p,q$ are both distinct primes. then

$a=p^2 q ,b=q, c=q^3$

To get minimum value put $p=3,q=2$ .it will give $a=18,b=2,c=8$

Solomon Olayta
Nov 28, 2016

We consider the smallest squares such as 1,4 and 9.To satisfy the condition, we form the numbers 2(1), 2(4) and 2(9). Thus, the smallest numbers are 2,8 and 18. Hence we get 2+8+18=28.

Geoff Pilling
Nov 22, 2016

If we enumerate all the pairs of positive integers < 25 whose product is a square and eliminate squares themselves we get:

2,8
2,18
3,12
5,20
6,24
8,18

You can see that 2, 8, and 18 are the only ones that lead to a triad of three numbers with the condition described above.

Therefore, the smallest values that a, b and c can be are 2, 8, and 18.

$2+8+18 = \boxed{28}$

The second smallest is a tie between $a + b + c = 3 + 12 + 27 = 42$ and $a + b + c = 2 + 8 + 32 = 42$ . The next smallest are $a + b + c = 2 + 8 + 50 = 60$ and then $5 + 20 + 45 = 70$ .

Brian Charlesworth - 4 years, 6 months ago

The numbers have to be as small as possible. Their primefactors can not all have even powers. They have to have some prime factors in common with uneven powers. So for the even powers we have to choose 2, for the uneven powers we choose 1 (small numbers). One of the numbers can in itself be prime en must then reoccur in the other numbers. This adds up to:

a = p, b = pq², c = pr², with p, q, r all prime. Keeping the numbers small we can choose p = 2 (considering p = 1 would yield to a square value for a). choosing as small as possible numbers for q and r and they can not be equal to 1 or to each other gives q = 2 and r = 3. Therefore (a, b, c) = (2, 8, 18). Choosing other primes will give other larger solutions.

Kris Hauchecorne - 4 years, 6 months ago

Nice explanation, @Kris Hauchecorne

Geoff Pilling - 4 years, 6 months ago

Got a proof?

Pi Han Goh - 4 years, 6 months ago

Added above

Geoff Pilling - 4 years, 6 months ago

Squares from non squares

The answer is 28.

3 solutions

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