The least perfect square number which is divisible by 3, 4, 5, 6 and 8 is
The least perfect square number which is divisible by 3, 4, 5, 6 and 8 is
A900
B1600
C2500
D3600
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correct answer is: 3600
Explanation: To find the least perfect square divisible by 3, 4, 5, 6, and 8, we first determine the least common multiple (LCM) of these numbers. The prime factorizations are: - 3 = \(3^1\) - 4 = \(2^2\) - 5 = \(5^1\) - 6 = \(2^1 \times 3^1\) - 8 = \(2^3\) The LCM is obtained by taking the highest powers of all primes: - \(2^3\) - \(3^1\) - \(5^1\) Thus, LCM = \(2^3 \times 3^1 \times 5^1 = 120\). To make this a perfect square, all exponents in its prime factorization must be even. Therefore, we multiply by \(2^1 \times 3^1 \times 5^1\) to get: - \(2^4 \times 3^2 \times 5^2 = 3600\). This is the smallest perfect square divisible by all the given numbers.
- Option a (900): Not divisible by 8 (\(2^3\)).
- Option b (1600): Not divisible by 3 or 6.
- Option c (2500): Not divisible by 3, 6, or 8.
- Option d (3600): Divisible by all and is a perfect square.
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