   SEARCH HOME Math Central Quandaries & Queries  Question from Peter, a student: What is the least possible positive integer-value of n such that square root(18*n*34) is an integer? Hi Peter,

Let me ask a different question.

Is the square root of 15 × 48 an integer?

You can answer this by looking at the prime factorization of 15 × 48.

15 × 48 = 3 × 5 × 2 × 2 × 2 × 2 × 3 = 24 × 32 × 51

The square root of 24 is 22 and the square root of 32 is 3 but the square root of 51 is not an integer so the square root of 15 × 48 = 24 × 32 × 51 is not an integer.

What is the least possible positive integer-value of n such that square root(15 × n × 48) is an integer?

Now I can see how to do this. In the prime factorization of 15 × n × 48 all the primes need to have an even power. 15 × n × 48 = 24 × 32 × 51 × n so if n = 5 then 15 × n × 48 = 24 × 32 × 52 and the square root of 15 × n × 48 is an integer.

Penny     Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.