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Question

Is 2352 a perfect square? If not, find the smallest number by which 2352 must be multiplied so that the product is a perfect square. Find the square root of the new number.


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Solution

Step 1: Find the prime factors of the given number :

The prime factors of 2352 are

2352=2×2×2×2×3×7×7=22×22×3×72

Since the 3 has no pair so 2352 cannot be a perfect square number.

Step 2: Find the smallest number:

From the prime factor of 2352 we can see that 3 has no pair.

So if we multiply 2352 with 3 it become a perfect square number

Therefore the new number will be 2352×3=7056

Step 3. Find square root of new number :

The prime factors of 7056 are

7056=2×2×2×2×3×3×7×77056=24×32×727056=24×32×727056=22×3×77056=84

Therefore the square root of the new number is 84

Hence, 2352 is not a perfect square number, so, 3 must be multiplied to make it a perfect square no and thus the new number is 7056. The square root of the new number is 84.


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