Mathematics
Assertion (A) : The smallest number by which 1323 may be multiplied so that the product is a perfect cube of 7.
Reason (R) : A given natural number is a perfect cube if in its prime factorization every prime occurs three times.
Both A and R are correct, and R is the correct explanation for A.
Both A and R are correct, and R is not the correct explanation for A.
A is true, but R is false.
A is false, but R is true.
Cube & Cube Roots
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Answer
Prime Factorization of 1323 = (3 x 3 x 3) x 7 x 7 = 33 x 72
On multiplying by 1323 by 7, we get :
⇒ 1323 x 7 = (3 x 3 x 3) x 7 x 7 x 7
⇒ 1323 x 7 = 33 x 73
⇒ 1323 x 7 = (3 x 7)3
⇒ 1323 x 7 = 213
∴ On multiplying 1323 by 7, it becomes a perfect cube.
So, assertion (A) is true.
In the prime factorization of a perfect cube, each prime must appear with an exponent that is a multiple of 3, it is not necessary that every prime number occurs only three times.
So, reason (R) is false.
Hence, option 3 is the correct option.
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