Find the smallest multiplier of 12748 such that the result is a perfect square.

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3187

7814

4577

1842

answer is A.

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Detailed Solution

Concept- The numbers from the option are arranged in ascending order as follows: 1842, 3187, 4577, and 7814. The lowest number, as can be seen, is 1842. Assuming that x and y are equal, we multiply them and then look for the prime factorization of the resulting product,

.The product is a perfect square if all the prime components in prime factorization appear an even number of times; otherwise, we move on to the following least numbers: 3187, 4577, and 7814.
If a composite number X has n prime factors,

, then

is its prime factorization.
A number with an integer square root is referred to as a perfect square. The prime factorization of

will have prime factors

happening half as often as they did for X, therefore we may calculate the square root of the number X if all of its prime factors

appear in an even number of instances.
We also know that all of X's prime factors will be found under

, which equals

. Therefore, by dividing X by all the primes less than

, we may determine a number's primality.
Assuming that

, determine x's prime factors. Given that 12748 is an even number, it must be divisible by 2, as is evident. It can be written as

as a result. Because 6374 is an even number, it may also be divided by 2, making

. We must now decide if 3187 is a prime number or not.

is the square that we have. In order to get the prime numbers that are smaller than

, we divide 3187 by each of the following: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, and 53. We find that none of these primes can divide 3187 precisely, demonstrating its primality. The prime factorization can therefore be used to write 12748 as

.
Let's determine the factorization of the option with the fewest possible values, 1842. We've got

We must determine whether 307 is a prime number or not.

; we have. By determining whether 307 can be divided by 2, 3, 5, 7, 11, 13, and 17, we may say that it is a prime number. Let's now assume that

and that the prime factorization of the

product is

. Those are

It is clear that none of the prime factors occur with an even frequency. Because of this, option 4 is untrue and the product is not a square perfectly. We now go to option A's third-to-last number, 3187. Since 3187 is already known to be a prime number, prime factorization will not occur. We use

and calculate the prime factorization of the

product. We have

We can observe that there are an equal number of occurrences of the prime factors 2 and 3187. The product's square root, which is an integer, is

.
Therefore, the result of multiplying 12748 by 3187 will be a perfect square.
Hence, option 1 is correct.

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