A reliable method
Translate equal groups to common divisors and repeated schedules to common multiples. For a remainder, use n = dq + r with 0 ≤ r < d.
The common trap
The greatest common divisor and least common multiple answer different questions. A larger number is not necessarily a multiple of the smaller one.
Check the conclusion
Verify the result by division. For a proposed minimum, check why every prime power is necessary; finding one working integer does not establish that it is the smallest.
Contrast the cases
Eighteen pens and thirty pads can make six identical kits, each holding three pens and five pads. Here the number of kits must divide both counts. Maintenance every eighteen and thirty days instead coincides after ninety days: the time must be a multiple of both intervals. Reversing these relationships produces a plausible-looking number that answers the wrong question. To make 12t divisible by 18, reduce 12/18 to 2/3; t must supply the missing factor three.
Transfer and explain
Transfer: compare gcd(24,40)=8 with lcm(24,40)=120, then find the smallest positive t making 24t divisible by 40: t=5. State why none of these answers is interchangeable.
A distributor has 45 pens and 75 notebooks. It makes the greatest possible number of identical kits, with every item used. How many kits does it make?
- The kit count must divide both item counts.
- The common factor of 15 and 25 is 5, and both counts include a factor 3.
- The greatest kit count is 15.
Compare the other answer choices
Returns pens per kit instead of number of kits.
Returns notebooks per kit instead of number of kits.
Assumes the smaller count divides the larger.
Uses a common multiple instead of a common divisor.
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.
GMAT exam contentSiyavula Grade 10 mathematics foundation reference