Question

the least number which is a perfect square and has 60 as a factor is

Answer 1

If your number must be an integer, you would need to solve for x:

60x=p2,{x,p}∈Z60x=p2,{x,p}∈Z

Note that 60=22⋅3⋅5.60=22⋅3⋅5.The prime factorization of a perfect square can be recognized by all exponents having even parity. In this case, 2 already does but 3 and 5 don’t; multiplying by 15 (3×5)15 (3×5) produces 900=302=22⋅32⋅52.900=302=22⋅32⋅52.

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Norman Chung

Answered Apr 30

If you consider 0^2 as a perfect square, then the lowest number that has to he multiplied to 60 to make a perfect square is zero.

If you want the perfect square of a positive integer and can use fractions, you could multiply 60 by 1/60 to get 1^2 or 1.

Lastly, if you want a perfect square that is greater than 60 AND where you multiply by an integer to get to that perfect square, we can start by prime factoring 60:

60 = 2^2 x 3 x 5

If you want a perfect square, you want every factor of that number to have an even power. Thus, you multiply 60 by 15 (3 x 5) to get all the powers even. This equals 900 or 30^2.

Answer 2

The perfect square is a multiple of 60.

Let's factorize 60 by LCM.

2 | 60

2 | 30

5 | 15

3 | 3

| 1

60 = 2² x 5 x 3

2² x 5 x 3 multiplied by 5 x 3 gives the least perfect square multiple of 60.

2² x 5 x 3 x 5 x 3 = 2² x 5² x 3² = (2 x 5 x 3)² = 30² = 900

∴ 900 is the answer.

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