A logistics manager is packing identical relief kits into crates. When she attempts to pac… — State PSC Exam

A logistics manager is packing identical relief kits into crates. When she attempts to pack them in equal batches of $16$ , $24$ , $30$ , or $36$ kits per crate, there are always exactly $8$ kits left over. However, if she packs them in batches of exactly $19$ kits per crate, there are zero kits left over. What is the least possible total number of relief kits she could be packing?

Cevap: 2888 kits

Cevap

The least possible total number of relief kits is 2888.

The correct answer is derived by finding the general form of a number that leaves a remainder of 8 when divided by 16, 24, 30, and 36. This form is

720k + 8

. By applying the final condition that the total number must be exactly divisible by 19, we find the smallest valid multiplier is

k=4

. Substituting this back gives

720(4) + 8 = 2888

Adım Adım Çözüm

Establish the relationship for the total number of kits based on the remainders.

The number of kits,

N

, leaves a remainder of

8

when divided by

16

24

30

, and

36

. Thus,

N = \text{LCM}(16, 24, 30, 36) \times k + 8

Any number that leaves the same remainder when divided by multiple divisors can be expressed as a multiple of their least common multiple plus that remainder.

Calculate the least common multiple (LCM) of

16

24

30

, and

36

The prime factorizations are

16 = 2^4

24 = 2^3 \times 3

30 = 2 \times 3 \times 5

, and

36 = 2^2 \times 3^2

. The LCM is the product of the highest powers:

2^4 \times 3^2 \times 5 = 720

The LCM is required to find the base repeating cycle for the division condition.

Formulate the exact divisibility condition.

Substitute the LCM into the equation to get

N = 720k + 8

. The problem states

N

is exactly divisible by

19

, so

(720k + 8) \equiv 0 \pmod{19}

This applies the second constraint of the problem to find the specific multiplier

k

Simplify the modular arithmetic equation to solve for

k

Divide

720

19

to find the remainder:

720 = 19 \times 37 + 17

. Substitute

17

for

720

to get

(17k + 8) \equiv 0 \pmod{19}

. This can be written as

(-2k + 8) \equiv 0 \pmod{19}

Simplifying large numbers using modulo properties makes finding the integer

k

manageable.

Find the smallest positive integer

k

that satisfies the equation.

Solving

-2k + 8 = 0

yields

2k = 8

, which means

k = 4

. Checking:

17(4) + 8 = 68 + 8 = 76

, and

76 \div 19 = 4

, which is exactly divisible.

Finding the smallest valid

k

ensures we calculate the least possible total number of kits.

Calculate the final total number of kits.

N = 720(4) + 8 = 2880 + 8 = 2888

Substituting

k = 4

back into the original formula for

N

provides the final numerical answer.

Anahtar Kavram

Finding a specific numerical value based on multiple simultaneous divisibility and remainder conditions using Least Common Multiple (LCM) and modular arithmetic.

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