Lcm may be the square

Number Theory Level pending

Let $\text{lcm}(a,b)$ denote the least common multiple of $a$ and $b$ . If the sum of all positive integers $x$ such that $x \leq 100$ and $\text{lcm}(16,x) = 16x$ is $K^2$ for $K>0$ then find $K$ .

The answer is 50.

1 solution

Ramesh Perumal
Jul 29, 2019

The key idea is to note that the condition holds if and only if x is odd. We present one of many arguments for why this is true: One can show that for positive integers a,b, gcd(a,b) · lcm(a,b) = ab. From this, it follows that gcd(16,x) = 1, so x must be odd. We thus need to compute the sum of the odd numbers less than 100, which is K^2= 50^2 = 2500. In a general form, the fact key fact for the above explanation is that gcd(a,b) = 1 if and only if lcm(a,b) = ab.

Lcm may be the square

The answer is 50.

1 solution

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