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This problem trains for: AMC-8, AMC-10.
The sum of two positive integers is:
The lest common multiple of the two integers is:
What is the positive difference of the two integers?
Factor the lcm into primes:
Since the sum of the numbers is even, the two numbers must be either both even or both odd. Since the lcm is even, at least one of the numbers must be even - therefore, they are both even. If both numbers are even, then their gcd is even and can be written as:
Therefore, the numbers can be written:
and their sum:
with both a and b odd and coprime.
Therefore, from the initial assertion:
Since a and b are both odd, coprime and, at the same time, are formed by factors from the lcm: