Subject: squares and cubes.

Name: Harman Chaudhry
Who is asking: Student
Level: Secondary

Question:
Which is the smallest 10-digit number to be five times the cube of one number and also three times the fifth power of another? Thank You.

You need to find a and b so that

Since 5 divides the left side, 5 must also divide the right side. Since 5 does not divide 3 it must divide b⁵. But 5 is a prime so if 5 divided b⁵ it must also divide b. Likewise 3 divides the right side so it follows that 3 divides a. Thus the left side must contain 5 3³ and the right side must contain 3 5⁵.

To make these equal we need four more 5's on the left side and two more 3's on the right side. That is

But 3³ 5⁴ is not a cube and 5⁵ 3³ is not a fifth power.

First fix up the 5's. Multiply both side by 5^k so that, on the left 5^4+k is a cube and, on the right 5^5+k is a fifth power. In a similar fashion fix the 3's

When you are done check to see that the number you have has 10 digits.