Question

17 × 17 × 18 × 18 × 19 × 19 shall be fully divisible by which of the following number?

• A. 971
• B. 173
• C. 646
• D. 233
• E. 5813

Hint:

Express the number in prime factors to check divisibility easily.

Answer 1

The Correct Option is C

Step 1: Understanding the Expression:
Expression = \(17^2 \times 18^2 \times 19^2 = (17 \times 18 \times 19)^2\)

Step 2: Calculating the Product:
\(17 \times 18 = 306\)
\(306 \times 19 = 5814\)
So, expression = \(5814^2\)

Step 3: Checking Divisibility:
\(5814 = 2 \times 2907 = 2 \times 3 \times 969 = 2 \times 3 \times 3 \times 323 = 2 \times 3^2 \times 17 \times 19\)
So, \(5814^2 = 2^2 \times 3^4 \times 17^2 \times 19^2\)
Now, 646 = \(2 \times 17 \times 19\)
Since the expression has \(2^2\), \(17^2\), and \(19^2\), it is divisible by \(2 \times 17 \times 19 = 646\).