Find the smallest number by which 3872 is to be
(i) divided to obtain a perfect Square
(ii) multiplied to obtain a perfect Cube.
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Find the smallest number by which 3872 is to be
(i) divided to obtain a perfect Square
(ii) multiplied to obtain a perfect Cube.
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Question
Find the smallest number by which 3872 is to be
(i) divided to obtain a perfect square
(ii) multiplied to obtain a perfect cube.
Answer
(i) First we must do prime factorization for 3872 to obtain the product of primes.
∴ 3872 = 2×2×2×2×2×11×11
Now we will pair the product of primes to check whether 3872 is a perfect square.
(2×2) × (2×2) × 2 × (11×11)
∴ 3872 is not a perfect square since '2' doesn't have a pair.
∴ 2 must be divided to 3872 to obtain a perfect square.
3872 ÷ 2 = 1936
Let's check if 1936 is a perfect square. First we will find the prime products.
∴ 1936 = 2×2×2×2×11×11
Now we will pair the product of primes to check whether 1936 is a perfect square.
(2×2) × (2×2) × (11×11)
∴ 1936 is a perfect square since all numbers have a pair.
∴ 2 must be divided by 3872 to obtain a perfect square.
(ii) First we must do prime factorization for 3872 to obtain the product of primes.
∴ 3872 = 2×2×2×2×2×11×11
Now we will group the product of primes with 3 in each group to check whether 3872 is a perfect square.
2 × 2 × (2×2×2) × 11 × 11
∴ 3872 is not a perfect cube since 2 and 11 are don't have a group.
∴ 2×11 = 22
∴ 22 must be multiplied to 3872 to obtain a perfect cube.
3872 × 22 = 85184
Let's check if 85184 is a perfect cube. First we will find the prime products.
∴ 85184 = 2×2×2×2×2×2××11×11×11
Now we will group the product of primes with 3 in each group to check whether 85184 is a perfect square.
(2×2×2) × (2×2×2) × (11×11×11)
∴ 85184 is a perfect cube since all numbers have a groups.
∴ 22 must be multiplied to 3872 to obtain a perfect square.
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