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. Hi Harman, You need to find a and b so that 5 a3 = 3 b5 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 5 (33 54) = 3 (55 33) 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 5k so that, on the left 54+k is a cube and, on the right 55+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. Cheers, Penny Go to Math Central