An inventory manager is organizing items into storage boxes. When the items are packed int… — GMAT

An inventory manager is organizing items into storage boxes. When the items are packed into boxes of $24$ , $36$ , or $45$ items each, there are always $11$ items remaining unpacked. If the total number of items is between $700$ and $1,000$ , how many items are there in total?

Answer: 731

Answer

731

The number of items

N

satisfies

N \equiv 11 \pmod{24}

N \equiv 11 \pmod{36}

, and

N \equiv 11 \pmod{45}

. Consequently,

N - 11

is a multiple of

\text{LCM}(24, 36, 45)

. The prime factorizations are

24 = 2^3 \times 3

36 = 2^2 \times 3^2

, and

45 = 3^2 \times 5

. The LCM is

2^3 \times 3^2 \times 5 = 360

. Thus,

N = 360k + 11

for some positive integer

k

. Testing values of

k

within the range

700 < N < 1,000

yields

k = 2

, giving

N = 360(2) + 11 = 731

Step-by-Step Solution

Find the Least Common Multiple (LCM) of the divisors 24, 36, and 45.

LCM(24, 36, 45) = 360

Any number that leaves a constant remainder when divided by a set of numbers must be a constant remainder greater than a common multiple of those numbers.

Formulate the general equation for the total number of items N.

N = 360k + 11

Adding the common remainder 11 to any multiple of the LCM yields a valid total number of items.

Evaluate integer values of k to satisfy the constraint 700 < N < 1,000.

k = 2 gives N = 731

k = 1 gives 371, which is too small, and k = 3 gives 1,091, which exceeds 1,000.

Key Concept

Least Common Multiple (LCM) with Constant Remainder

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