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1.Suppose t is a positive integer such that lcm[12,t]^3=(12t)^2 . What is the smallest possible value for t?

 

2. The value b^n has both b and n as positive integers less than or equal to 15. What is the greatest number of positive factors b^n can have?

 Jun 4, 2018
 #1
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1.

 lcm[12,t]^3=(12t)^2

Lcm[12^3, t^3] = 144t^2

Lcm[1,728, t^3] = 144t^2

The smallest positive t = 18, so that:

Lcm[12^3, 18^3] = 12^2 * 18^2

 Jun 5, 2018
 #2
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12^15 = 15407021574586368 = 2^30×3^15 (45 prime factors, 2 distinct). It also has 496 divisors.

 Jun 5, 2018

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