The product of two numbers is 4107 and their HCF is 37. What is the larger of the two numbers?

Write the numbers as $37a$ and $37b$, where a and $b$ have no common factor. Their product is $37^2\times ab=4107$, and since $37^2=1369$ we get $ab=4107÷1369=3$. The only co-prime choice is $a=1$ and $b=3$, so the numbers are 37 and 111. Check: $37\times 111=4107$ and HCF(37, 111) = 37. The larger is 111; answering 37 reports the smaller number, and $148=4\times 37$ does not divide 4107 at all.