# GMAT 700 Level Quant Question 35

#### Number Properties Practice | Remainders | Least Common Multiple

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?

1. 2464
2. 2477
3. 2451
4. 616
5. 603

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### Explanatory Answer | GMAT Number Properties

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 = 22 × 11
56 = 23 × 7
32 = 25

Step 2 of Computing LCM: LCM is the product of highest power of all primes.
LCM (44, 56, 32) = 25 × 7 × 11 = 2464
Therefore, N + 13 = 2464
Hence, N = 2464 – 13 = 2451

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