A reliable method
Write a remainder statement as dividend = divisor × integer + remainder, with the remainder between zero and one less than the positive divisor. For common factors, list or factor both integers.
The common trap
A number can leave the same remainder after two different divisions without being divisible by either divisor. A prime factor and a positive factor are not the same count.
Make the connection
Use a short list to test a general claim. When the question asks for all possible integers, check both the algebraic condition and the stated integer restriction.
Contrast the cases
Compare two decisions: 18 and 30 objects form at most 6 identical groups, so use gcd. Events repeating every 18 and 30 days next coincide after 90 days, so use lcm. Both questions involve the same pair of numbers, but “divides both” and “multiple of both” are opposite relationships. A remainder is a third relationship: 32 = 5(6)+2, so remainder 2 is not the quotient 6.
Transfer and explain
Transfer: find gcd(24,36), lcm(24,36), and the remainder of 37 divided by 6. Results: 12, 72, 1. Explain why one answer cannot serve all three requests.
An integer n equals 50. What remainder does 3n + 1 leave when divided by 6?
- n leaves remainder 2 on division by 6.
- Therefore 3n + 1 has the same remainder as 3(2) + 1 = 7.
- Reduce 7 modulo 6: the remainder is 1.
Compare the other answer choices
Forgets the final +1 before reducing.
Uses the remainder of n rather than 3n+1.
Keeps only the multiplier of n.
Squares the original remainder instead of applying 3n+1.
Try it, then explain it.
Complete a short set without looking at the example. After each answer, explain which operation preserved the relationship and why the result makes sense.
Sources & authorship
Original Slope explanations and examples. References support the concepts; they are not the source of copied questions. Independent educator review is pending.
GRE Quantitative Reasoning overviewSiyavula Grade 10 mathematics foundation reference