A whole family of GRE word problems describes a number by the remainders it leaves. "A number leaves remainder 2 when divided by 3 and by 4" means it has the form (multiple of 12) + 2, because 12 is the LCM of 3 and 4 and the shared remainder rides along. The general strategy: when the same remainder r appears for several divisors, the answer is LCM(divisors)·k + r. When remainders differ, look for a common "deficit" — "remainder 4 on division by 5 and remainder 6 on division by 7" both fall exactly 1 short of the divisor, so the number is LCM(5,7)·k − 1 = 35k − 1. If no clean pattern appears, list the smallest candidates for one condition and test them against the other. These questions look like puzzles but reduce to the division identity plus an LCM.
✅ Solved examples
1. Find the smallest number greater than 1 that leaves remainder 1 when divided by 2, 3 and 4.
Same remainder 1 for all, so the number is LCM(2,3,4)·k + 1 = 12k + 1. The smallest above 1 is 12·1 + 1 = 13.
2. A number leaves remainder 3 on division by 5 and remainder 3 on division by 7. Find the smallest positive such number.
Shared remainder 3, so it is LCM(5,7)·k + 3 = 35k + 3. Smallest positive: k = 0 gives 3 (which indeed leaves remainder 3 for both). Next is 38.
3. What is the smallest number that leaves remainder 4 on division by 5 and remainder 6 on division by 7?
Each remainder is 1 less than its divisor (5−1, 7−1), so the number is LCM(5,7)·k − 1 = 35k − 1. Smallest positive: 35·1 − 1 = 34. (Check: 34 = 6·5+4 and 34 = 4·7+6.)
4. Quantitative Comparison. n leaves remainder 2 on division by 6. Quantity A: the remainder when n is divided by 3. Quantity B: 2. Which is greater?
n = 6k + 2 = 3(2k) + 2, so dividing by 3 also leaves remainder 2. Quantity A = 2 = Quantity B; they are equal.
✏️ Practice — try these, take hints as needed
1. Smallest number > 1 leaving remainder 1 on division by 3 and 5?
Same remainder 1.
LCM(3,5)·k + 1.
15k + 1.
16
2. Smallest positive number leaving remainder 2 on division by 4 and 6?
LCM(4,6) = 12.
12k + 2.
Smallest positive.
2 (next is 14)
3. Smallest number leaving remainder 5 on division by 6 and remainder 7 on division by 8?
Each is 1 short.
LCM(6,8) = 24.
24k − 1.
23
4. If n leaves remainder 3 on division by 10, what remainder does it leave on division by 5?
n = 10k + 3.
10k is a multiple of 5.
Remainder from the 3.
3
5. Smallest number > 2 leaving remainder 2 on division by 3, 4 and 5?
Same remainder 2.
LCM(3,4,5) = 60.
60k + 2.
62
📝 Topic test — 8 questions
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