Problem
Product equals GCD times LCM
Every pair of numbers carries a product, a GCD and an LCM. The three cases give product 216 with LCM 72, product 63 with GCD 3, and GCD 60 with LCM 240. The missing one has to be found. Answer all three.
Reasoning · Grade 5-1 Greatest Common Divisor and Least Common Multiple
Your answer
Watch out Finding the pair looks necessary, but the relation among the three is enough.
How to solve
Strategy Solve an Easier Related Problem — Rather than attacking the three parts one at a time, I first try a pair of numbers I can see all the way through — 12 and 18 — and watch what happens to the product, the greatest common divisor and the least common multiple. That small case reveals the rule (product = GCD × LCM) and, better, shows why it is true: writing the two numbers as the GCD times two numbers that share no factor makes both sides come out the same. Once the rule is in hand, each of the three parts is a one-step 'work backwards': whichever of the three quantities is missing, it is either a division or a multiplication away. At the end I list the actual number pairs to make sure the answers really are achievable.
1STEP 1
Find the rule on a pair I can see completely
The product equals GCD times LCM.
A = a × G, B = b × G → LCM = a × b × G, A × B = a × b × G × G = G × LCM
The greatest common divisor is what the two numbers share; the least common multiple is everything either of them has. Multiply them and you have counted each shared factor once and each unshared factor once — which is exactly what multiplying the two numbers does.
6.NS.B.4Solve An Easier Related ProblemFor any two numbers the greatest common divisor multiplied by the least common multiple equals their product.
Why?
Each prime appears a fixed number of times in each number, and the smaller count plus the larger count is the two counts added together.
🧱Every number has one prime factorizationKeep splitting a number into factors and you always land on the same bag of primes, no matter which split you tried first.
Why?
Testing the claim on a pair whose divisors can all be listed would expose it at once if it were false, and it survives.
🧱One counterexample refutes an always-claimA claim about every case dies the moment you exhibit one case where it fails, and exhibiting is easier than arguing.
2STEP 2
(1) The product and the LCM are given, so divide
(1) divides the product by the LCM: 3.
216 = (GCD) × 72 → GCD = 216 ÷ 72 = 3
This is an unknown-factor division: you know the product and one factor, so the other factor comes from dividing — the same move as filling in □ in 216 = □ × 72.
5.NBT.B.6Work Backwards3STEP 3
(2) The product and the GCD are given, so divide the other way
(2) divides the product by the GCD: 21.
63 = 3×(LCM) → LCM = 63 ÷ 3 = 21
The same rule serves both directions — which of the two you divide by depends only on which one you were handed.
5.NBT.B.6Work Backwards4STEP 4
(3) The GCD and the LCM are given, so multiply
(3) multiplies the two: 14400.
(product) = 60 × 240 = 14400
60 × 240 is 6 × 24 with two zeros attached: 144 becomes 14400.
5.NBT.B.5Work Backwards5STEP 5
Check that real number pairs exist for all three answers
Real pairs exist for all three answers.
(3,72): 3 × 72=216, LCM=72, GCD=3; (3,21): 3 × 21=63, GCD=3, LCM=21; (60,240): GCD=60, LCM=240, product=14400
Listing factor pairs of the product and checking each one is slow but completely reliable, and it turns an abstract identity into something you can point at.
4.OA.B.4Make A Systematic ListAnswer
3, 21, 14400
216 ÷ 72 = 3, 63 ÷ 3 = 21, 60 × 240 = 14400
All three answers pass the basic size tests. In (1) the greatest common divisor must divide the least common multiple, and 3 does divide 72; it must also be no bigger than either number, and 3 is comfortably small. In (2) the least common multiple must be a multiple of the greatest common divisor and must not exceed the product, and 21 is a multiple of 3 sitting well below 63. In (3) the product must be at least as big as the least common multiple, and 14400 is far larger than 240 — which is right, because the GCD 60 is large, meaning the two numbers overlap heavily. Every answer was also checked against an actual pair of numbers: (3,72) or (9,24) for part (1), (3,21) for part (2), and (60,240) for part (3).
Takeaway
The greatest common divisor counts what two numbers share and the least common multiple counts everything they have between them, so multiplying those two always gives back the product of the numbers.
- Find the rule on a pair I can see completely
- (1) The product and the LCM are given, so divide
- (2) The product and the GCD are given, so divide the other way
- (3) The GCD and the LCM are given, so multiply
- Check that real number pairs exist for all three answers