A number theory problem by Trung Le

Number Theory Level 3

Two number less than 1000 have their greatest common divisor 13 and least common multiple 2015. Find their sum.

The answer is 468.

1 solution

Kay Xspre
Oct 30, 2015

We use the solution of $gcd (a,b)\times lcm (a,b) = a\times b$ . The product of $ab$ is then equal to $2015 \times 13 = 26195$ . As both $a, b$ has GCD of 13, we can write as $26195 = (13\times31)(13\times5)$ . As 31 and 5 is prime, we can then conclude that $(a,b) = (403,65)$ and $a+b = 468$

lcm/gcd = 155; 155 = 31 x 5; since gcd(a,b) = 13 => (a,b) = ((31 x gcd), (5 x gcd) => (a,b) = ((31 x 13), (5 x 13)) = (403, 65) therefore, sum (a,b) = sum(403,65) = 468

Michael Fitzgerald - 3 years, 9 months ago

A number theory problem by Trung Le

The answer is 468.

1 solution

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