Question

If M, A and T ate distinct positive integers such that M A T = 1947, then which of the following is the maximum possible value of M + A + T?

• A. 189
• B. 649
• C. 653
• D. 1949

Answer 1

The Correct Option is C

To find the maximum possible value of \( M + A + T \) given that \( M \), \( A \), and \( T \) are distinct positive integers and their product \( M \times A \times T = 1947 \), we need to break down the steps and logic involved:

1. 1947 is a product of three distinct positive integers. First, we need to factorize 1947 to understand its prime components.

   The prime factorization of \( 1947 \) is determined as follows:

   • \( 1947 \div 3 = 649 \) (since the sum of digits, 1 + 9 + 4 + 7 = 21, is divisible by 3)
   • \( 649 \) is a prime number (checked by ensuring it's not divisible by any prime number up to its square root, approximately 25.5).

   So, \( 1947 = 3 \times 649 \).

2. Since \( 1947 \) has only two prime factors, \( 3 \) and \( 649 \), and they need to be divided among three distinct integers, the smallest integer not used yet is \( 1 \).
3. This suggests possible distinct integers \( M = 1 \), \( A = 3 \), \( T = 649 \).
4. Now, calculate their sum: \( M + A + T = 1 + 3 + 649 = 653 \).

Therefore, the maximum possible value of \( M + A + T \) is 653.

Among the given options, 653 is indeed one of them.