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

Hint:

Find the LCM first, then multiply by the smallest factor needed to make every prime's power even.

Answer 1

The Correct Option is B

To find the least number which is a perfect square and divisible by each of the numbers 14, 16, and 18, follow these steps:

1. First, find the Least Common Multiple (LCM) of the numbers 14, 16, and 18. 
2. The prime factorization of each number is as follows:
   • \(14 = 2 \times 7\)
   • \(16 = 2^4\)
   • \(18 = 2 \times 3^2\)
3. To find the LCM, select the highest power of each prime number available:
   • For prime number 2, the highest power is \(2^4\).
   • For prime number 3, the highest power is \(3^2\).
   • For prime number 7, the highest power is \(7^1\).
4. Therefore, the LCM is: \(LCM = 2^4 \times 3^2 \times 7 = 16 \times 9 \times 7 = 1008\)
5. To make this number a perfect square, each prime factor's exponent must be even. Check the exponents in the prime factorization of 1008:
   • \(1008 = 2^4 \times 3^2 \times 7^1\)
   • The exponent of 7 is odd. We need one more 7 to make it even, i.e., \(7^2\).
6. Thus, to make it a perfect square, multiply 1008 by 7: \(1008 \times 7 = 7056\)
7. Verify that 7056 is a perfect square:
   • \(7056 = 2^4 \times 3^2 \times 7^2\)
   • All exponents are now even, confirming it is a perfect square.

Therefore, the least number which is a perfect square and divisible by each of the numbers 14, 16, and 18 is 7056.

Answer 2

Step 1: Find the LCM of 14, 16 and 18.
Break each number into primes: \(14 = 2 \times 7\), \(16 = 2^4\), \(18 = 2 \times 3^2\). The LCM takes the highest power of each prime: \(2^4 \times 3^2 \times 7 = 1008\).

Step 2: Check if 1008 is a perfect square.
Write \(1008 = 2^4 \times 3^2 \times 7^1\). For a perfect square every prime's power must be even. Here 7 has power 1, which is odd, so 1008 is not a perfect square.

Step 3: Fix the odd power.
Multiply by one more 7 so its power becomes 2: \(1008 \times 7 = 2^4 \times 3^2 \times 7^2 = 7056\).

Final Answer:
Check: \(\sqrt{7056} = 84\), and 7056 is divisible by 14, 16 and 18. \[ \boxed{7056} \]