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30 May 2026, 09:54
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A stationery store received a shipment of notebooks. When the manager packs the notebooks into cartons that hold 8 notebooks each, 5 notebooks are left unpacked. If the same shipment is packed into cartons that hold 12 notebooks each, how many notebooks will be left unpacked?
(1) If the notebooks are packed into cartons that hold 6 notebooks each, 5 notebooks are left unpacked.
(2) If the notebooks are packed into cartons that hold 10 notebooks each, 5 notebooks are left unpacked.
(1) If the notebooks are packed into cartons that hold 6 notebooks each, 5 notebooks are left unpacked.
(2) If the notebooks are packed into cartons that hold 10 notebooks each, 5 notebooks are left unpacked.
30 May 2026, 09:54
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Official Solution:
A stationery store received a shipment of notebooks. When the manager packs the notebooks into cartons that hold 8 notebooks each, 5 notebooks are left unpacked. If the same shipment is packed into cartons that hold 12 notebooks each, how many notebooks will be left unpacked?
(1) If the notebooks are packed into cartons that hold 6 notebooks each, 5 notebooks are left unpacked.
We are given that the number of notebooks leaves 5 when divided by 8, and statement (1) says it also leaves 5 when divided by 6. So the total must be 5 more than a common multiple of 8 and 6. The least common multiple of 8 and 6 is 24, so the total is of the form
\(24k + 5\)
When such a number is divided by 12, the remainder is always 5. Sufficient.
(2) If the notebooks are packed into cartons that hold 10 notebooks each, 5 notebooks are left unpacked.
Here the total leaves 5 when divided by 8 and also 5 when divided by 10.
That means the total is 5 more than a common multiple of 8 and 10. The least common multiple of 8 and 10 is 40, so the total is of the form
\(40k + 5\)
For different values of \(k\), \(40k + 5\) can leave different remainders when divided by 12. For example:
If \(k = 1\), then \(40k + 5 = 45\), which leaves remainder 9.
If \(k = 2\), then \(40k + 5 = 85\), which leaves remainder 1.
Not sufficient.
Answer: A
A stationery store received a shipment of notebooks. When the manager packs the notebooks into cartons that hold 8 notebooks each, 5 notebooks are left unpacked. If the same shipment is packed into cartons that hold 12 notebooks each, how many notebooks will be left unpacked?
(1) If the notebooks are packed into cartons that hold 6 notebooks each, 5 notebooks are left unpacked.
We are given that the number of notebooks leaves 5 when divided by 8, and statement (1) says it also leaves 5 when divided by 6. So the total must be 5 more than a common multiple of 8 and 6. The least common multiple of 8 and 6 is 24, so the total is of the form
\(24k + 5\)
When such a number is divided by 12, the remainder is always 5. Sufficient.
(2) If the notebooks are packed into cartons that hold 10 notebooks each, 5 notebooks are left unpacked.
Here the total leaves 5 when divided by 8 and also 5 when divided by 10.
That means the total is 5 more than a common multiple of 8 and 10. The least common multiple of 8 and 10 is 40, so the total is of the form
\(40k + 5\)
For different values of \(k\), \(40k + 5\) can leave different remainders when divided by 12. For example:
If \(k = 1\), then \(40k + 5 = 45\), which leaves remainder 9.
If \(k = 2\), then \(40k + 5 = 85\), which leaves remainder 1.
Not sufficient.
Answer: A
