The given question is a GMAT hard math problem solving question from the topic number properties. It tests the concept of Remainders and Least common multiple.

Question 35: What is the least number that when divided by 44 leaves a remainder 31, when divided by 56 leaves a remainder 43, and when divided by 32 leaves a remainder 19?

- 2464
- 2477
- 2451
- 616
- 603

@ INR

Let N be the given number.

N leaves a remainder 31 when divided by 44, i.e., For \\frac{N}{44}), remainder is 31

So, N + 13 will be divisible by 44.

N leaves a remainder 43 when divided by 56, i.e., For \\frac{N}{56}), remainder is 43

So, N + 13 will be divisible by 56.

N leaves a remainder 19 when divided by 32, i.e., For \\frac{N}{32}), remainder is 19

So, N + 13 will be divisible by 32.

Hence, N + 13 will be divisible by 44, 56, 32. i.e., N + 13 is a multiple of 44, 56, 32.

The least value of (N + 13) is the LCM of 44, 56, and 32.

**Find the LCM of 44, 56, 32**

**Step 1 of Computing LCM**: Prime Factorize 44, 56, and 32

44 = 2^{2} × 11

56 = 2^{3} × 7

32 = 2^{5}

**Step 2 of Computing LCM**: LCM is the product of highest power of all primes.

LCM (44, 56, 32) = 2^{5} × 7 × 11 = 2464

Therefore, N + 13 = 2464

Hence, N = 2464 – 13 = **2451**

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