Grade 5 Factors and Multiples

What this problem type covers

Twenty-three problem types across five levels: recognizing and listing factors, finding the nth multiple, counting multiples in an interval, comparing factor counts among multiples, recognizing factor-multiple pairs, greatest common factors and all common factors, least common multiples and later common multiples, equal groups and largest square pieces, the smallest square made from rectangles, numbers in a range related as a factor or multiple, recovering a divisor from remainders, recovering a number from a GCF and LCM, repeated schedules and clock times, filling a digit to meet a divisibility rule, meshed gears, nested defined operations using factor counts, distributions constrained by common factors, garden lights and boxes stacked into a cube, inclusion-exclusion for numbers that are neither multiple, overlapping fountain cycles, pairs constrained by sum or a quotient clue with GCF and LCM, linked or independent missing digits under two divisibility rules, and range searches with the same remainder.

Example problems

One example from each difficulty band, with worked solutions.

These problems are generated automatically. The answers are derived as each problem is built and are checked by tests, but mistakes are still possible — please tell us if you find one.

1. 1 · ★1
   Choose every pair in which one number is a factor of the other.
   ① 30, 43　② 14, 42　③ 4, 8　④ 28, 48　⑤ 24, 33
   • ①③
   • ②②
   • ③②, ③, ④
   • ④①, ②, ③
   • ⑤②, ③
   Show solution

   ⑤②, ③

   1. Divide the larger number in each pair by the smaller.
   2. A remainder of 0 means the pair has a factor-multiple relationship.
   3. So the answer is ②, ③.
2. 2 · ★2
   From 9 through 160, find every number that is a factor or a multiple of 40.
   • ①10, 40, 80, 120, 160
   • ②10, 20, 80, 120, 160
   • ③10, 20, 40, 80, 120, 160
   • ④10, 20, 40, 120, 160
   • ⑤20, 40, 80, 120, 160
   Show solution

   ③10, 20, 40, 80, 120, 160

   1. First find the factors of 40 in the interval.
   2. Then find its multiples in the interval.
   3. Together they are 10, 20, 40, 80, 120, 160.
3. 3 · ★3
   220 leaves remainder 16, and 308 leaves remainder 2, when divided by the same number. Find the smallest possible divisor.
   • ①17
   • ②4
   • ③102
   • ④16
   • ⑤44
   Show solution

   ①17

   1. Subtract the remainders to get 204 and 306.
   2. The divisor must be their common factor and exceed both remainders.
   3. The possibilities are 17, 34, 51, 102, so the answer is 17.
4. 4 · ★4
   The smallest cube is stacked from soap boxes measuring 9 cm by 12 cm by 16 cm. How many boxes are needed?
   • ①144
   • ②108
   • ③37
   • ④1,728
   • ⑤1,727
   Show solution

   ④1,728

   1. The cube's side is the LCM of the three lengths, 144 cm.
   2. Find the block count in each direction and multiply.
   3. That gives 1,728 blocks.
5. 5 · ★5
   Two whole numbers add to 88. Their greatest common factor is 8 and their least common multiple is 224. Find the two numbers in increasing order.
   • ①24 and 64
   • ②16 and 72
   • ③32 and 56
   • ④8 and 224
   • ⑤32 and 224
   Show solution

   ③32 and 56

   1. Write both numbers as multiples of the GCF, 8.
   2. The quotients are coprime, and their product is the LCM divided by the GCF.
   3. Checking the sum gives 32 and 56.

Where people usually go wrong

• Treating 1 as the first multiple. The first positive multiple of a number is the number itself.
• Stopping the factor list after the smaller half of a factor pair. Every quotient in the pair is a factor too.
• Using the greatest common factor where a repeating schedule needs the least common multiple.
• Counting the lower endpoint of a range without checking whether it is actually a multiple.
• Subtracting only one remainder when two divisions have different remainders.
• Using the product of two numbers as their LCM even when the numbers have a common factor.
• Checking only the even final digit for divisibility by 6 and skipping the digit-sum test for 3.
• Adding two counts in inclusion-exclusion without removing the common multiples counted twice.
• Counting a spray-rest cycle as entirely active instead of intersecting the two active intervals.