Find two natural numbers whose difference is 66 and the least common multiple is 360.

For any two numbers aaa and bbb: a×b=LCM(a,b)×GCD(a,b) Let the GCD be g. Then we can write: a=g×x,b=g×y where xxx and yyy are coprime numbers (i.e., their GCD = 1). a−b=66a - b = 66 LCM(a,b)=360 So: gxy=360 and g(x−y)=66 From g (x−y)=66, we can express: x−y=66/g Since x and y are coprime and x>y, both must be integers. Now, g must be a factor of 66. Possible g: 1, 2, 3, 6, 11, 22, 33, 66. We also have gxy=360xy = 360g We’ll test values: Case g = 6 Then x−y=66/6=11 and xy=360/6=60. We need two coprime numbers whose product = 60 and difference = 11. Try pairs: • (15, 4) → 15×4=60, difference = 11 and GCD(15,4)=1 Perfect. a=g×x=6×15=90 b=g×y=6×4=24 Correct Answer: The numbers are 90 and 24.