2023/015) The sum of two numbers is 253 and their L.C.M. is 644. What are the numbers?

 Let the 2 numbers be ma and mb where GCD(a,b) = 1 and hence m is GCD of the 2 numbers and a > b

Sum = m(a+b) = 253 and LCM = mab = 644

As GCD(a,b) = 1 so gcd(a+b, ab ) =1 so m is gcd of 253 and 644

253 = 23 * 11

644 = 23 * 28

So the GCD = 23

The numbers are 23a and 23b 

$a+ b = 11\cdots(1)$

$ab = 28\cdots(2)$

So $(a-b)^2 = (a+b)^2 - 4ab = 11^2 - 4 * 28 = 9$

Or $a-b= 3\cdots(3$

Adding (1) and (3) we get $2a =14$ or $a= 7$ and putting in 1) we get b = 3  

Hence 2 numbers are 23 * 4 = 92 and 23 * 7 = 161