GCF, LCM, and a Generalization That Does Not Extend
During a number-theory lesson, a student computes GCF(84,120)=12 and LCM(84,120)=840, then notes that 12×840=10080=84×120. The student writes a conjecture: 'For any set of whole numbers, the GCF times the LCM equals the product of the numbers, so this must work for three numbers too.' You ask the class to test the conjecture on 18, 24, and 30. One group computes GCF=6 and LCM=360, multiplies to get 2160, but 18×24×30=12960, so the conjecture fails. The student responds that the group must have made an arithmetic mistake, because 'patterns that work for two cases should work for all.'
1. Verify the two-number identity for 84 and 120 using prime factorizations, and explain why it holds in general for two numbers.
Read the worked answer
Factor: 84=2^2·3·7 and 120=2^3·3·5. The GCF uses the smaller exponent of each prime: 2^2·3=12. The LCM uses the larger exponent: 2^3·3·5·7=840. Their product uses, for each prime, the sum of the smaller and larger exponents, which is exactly the exponent in 84·120. So 12·840=10080=84·120. The identity is a consequence of min+max equaling the total for two exponents, not a coincidence of the two cases. It does not automatically extend to three numbers because with three exponents the middle value is neither min nor max.
2. Test the three-number conjecture with 18, 24, and 30. What valid relationship replaces the false generalization, and what does this show about generalizing from cases?
Read the worked answer
The group is correct. For 18=2·3^2, 24=2^3·3, and 30=2·3·5, the GCF is 2·3=6 and the LCM is 2^3·3^2·5=360, so GCF·LCM=2160, while 18·24·30=12960. The missing factor is 6. For each prime, the product of the three numbers uses the sum of all three exponents, but GCF·LCM uses only the smallest and largest, leaving the middle exponent. Here the middle factors are 2^1 and 3^1. A valid generalization must account for those middle exponents; checking two cases, or even many cases, cannot establish a universal claim without a structural argument.