Question

The least number which is a perfect square and is divisible by each of the numbers 14, 16, 18 is

• A. 6048
• B. 7056
• C. 1008
• D. 2046

Answer 1

The Correct Option is B

To find the least number which is a perfect square and divisible by 14, 16, and 18, we need to follow these steps:

• Calculate the Least Common Multiple (LCM) of the numbers 14, 16, and 18. 
• Make the LCM a perfect square by adjusting its factors.

1. Find the LCM:

Prime factorize each number:

• 14 = \(2^1 \times 7^1\)
• 16 = \(2^4\)
• 18 = \(2^1 \times 3^2\)

The LCM is found by taking the highest power of all prime factors appearing in the factorizations.

• LCM = \(2^4 \times 3^2 \times 7^1 = 1008\)

2. Make the LCM a perfect square:

The number 1008 needs to be 'completed' to be a perfect square. The prime factorization of 1008 is:

• 1008 = \(2^4 \times 3^2 \times 7^1\)

To make it a perfect square, each prime's exponent must be even. Exponents of 2 and 3 are already even, but 7's exponent (1) is odd, so we multiply by 7.

\(1008 \times 7 = 7056\)

Prime factorization of 7056: \(2^4 \times 3^2 \times 7^2\). All exponents are now even, making 7056 a perfect square.

Therefore, the smallest perfect square divisible by 14, 16, and 18 is 7056.

Option	Value
A	6048
B	7056
C	1008
D	2046

Correct Answer: 7056