How to solve this Elementary Maths question?

Given that 1500k is a perfect square, write down the smallest possible value of k.How to solve this Elementary Maths question?
square root of 1500k.How to solve this Elementary Maths question?
Find the prime factorization of 1500.





1500 = 100 * 15 = 10^2 * 3 * 5 = 2^2 * 3^1 * 5^3





A perfect square will have all even exponents in the prime factorization. The smallest number you can multiply to make them even is 3 * 5 = 15





k = 15
1500 k = 100(15k) = the 100 is a square so (15k) must be as square works is k = 15
1500k = 100*(15k)





In order for 1500k to be a perfect square, both 100 and 15k must be perfect squares where 15k =/ 100





Since 100 is already a perfect square (ie 10^2) we need not worry about it





Now 15k must be a perfect square. The smallest multiple of 15 that is a perfect square is 15 (ie 15*15 = 225), therefore the smallest possible value for k is 15.





Hope that helped =D
on factorizing - 1500 = 2x2x5x5x5x3


so k=15 i.e. (5x3)
Write down 1500 as product of its prime factors (other than 1) :


1500 = 2 * 2 * 3 * 5 * 5 * 5


Now count how many times each prime factor appear in above product. For number to be a perfect square each factor should appear even number of times. We can see that factor 2 appears 2 times (even number) but 3 appears once (odd number) and 5 appears three times (also odd number). To make perfect square out of 1500, we should multiply it by 3 (to make factor 3 to appear twice) and by 5 (to make 5 appear four times in total). Therefore number


1500 k = 2 * 2 * 3 * 5 * 5 * 5 * k


is perfect square if k=3*5=15


The answer is k=15
1500k=100*15k


=100*15*15=10*15=150





there fore 15 is the answer

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