Prime-power form check
Before beginning, express 72 and 120 as products of prime powers. If either factorisation is incomplete, revisit the prime-factorisation lesson.
Learning goals
- Find HCF and LCM using prime-power representations.
- Use HCF(a,b) x LCM(a,b) = a x b for two positive integers.
- Choose HCF or LCM appropriately in contextual problems.
- Explain why the two-number product relation cannot be extended directly to three numbers.
Concept lesson
Factors versus multiples
A factor divides a number exactly; a multiple is produced by multiplying the number by an integer. HCF searches downward among common factors. LCM searches upward among common multiples.
Two questions hidden in the same prime factors
Prime factorisations show both what numbers share and what is needed to contain them completely. HCF keeps only the prime powers shared by every number. LCM collects enough of every prime to contain each number as a factor.
Highest Common Factor
The HCF is the greatest positive integer dividing every given number. In prime-power form, use only primes common to all numbers and choose the least exponent of each.
Least Common Multiple
The LCM is the least positive integer divisible by every given number. In prime-power form, include every prime appearing in any number and choose the greatest exponent of each.
Least for HCF, greatest for LCM
Compare exponents prime by prime. HCF is restricted by the smaller available exponent. LCM must be large enough to include the larger exponent. A two-row exponent table makes the choice visible.
Product relation for two positive integers
For two positive integers a and b: HCF(a,b) x LCM(a,b) = a x b. This relation can find a missing number, HCF or LCM when the other three quantities are known.
Why the product relation is not copied to three numbers
The identity HCF x LCM = product is guaranteed for two positive integers. A counterexample disproves its direct extension: for 2, 3 and 6, HCF = 1 and LCM = 6, so their product is 6, while 2 x 3 x 6 = 36.
Find the HCF of 72 and 120
Find HCF(72,120) using prime factorisation.
Find the LCM of 72 and 120
Find LCM(72,120) using the same prime factorisations.
Find a missing number using HCF and LCM
The HCF and LCM of two positive integers are 6 and 180. If one integer is 30, find the other.
Find HCF and LCM of three numbers
Find the HCF and LCM of 12, 18 and 30 using prime powers. Do not use the two-number product shortcut.
Select exponents for HCF and LCM
Given A = 2^4 x 3^2 x 7 and B = 2^2 x 3^3 x 5, complete an exponent table for primes 2, 3, 5 and 7. Select least common exponents for HCF and greatest available exponents for LCM.
Repair an incomplete LCM
A student finds LCM(18,40) as 2 x 3 = 6 because 2 is common and 3 occurs in 18. Identify every error, then calculate the correct LCM and verify divisibility by 18 and 40.
Test the three-number product claim
For 4, 6 and 12, calculate HCF x LCM and compare it with 4 x 6 x 12. State what this counterexample proves about directly extending the two-number relation.
HCF or LCM decision practice
Decide without calculating: (a) finding the greatest equal length for cutting two ribbons, (b) finding when two repeating alarms next ring together, (c) making the greatest number of identical groups. Explain the clue that signals HCF or LCM.
Choose before calculating
Use HCF when dividing quantities into the greatest possible equal units or groups. Use LCM when seeking the first or least value at which repeating cycles coincide. Identify the required structure before doing arithmetic.
Two runners on a circular path
Two runners complete a lap in 18 minutes and 12 minutes. Starting together in the same direction, find when they next meet at the starting point. Because the event repeats, calculate LCM(18,12).
One-mark direct calculation
Direct HCF or LCM questions may carry one mark. Write the complete selected answer in the board format, not only the option letter.
Two-mark product-relation pattern
State HCF x LCM = product of the two integers, substitute all known values, solve for the missing value, and verify if requested. Showing the relation and substitution protects method marks.
Decision strip
HCF: common primes, least powers, greatest equal grouping. LCM: all involved primes, greatest powers, least common occurrence. For exactly two positive integers, HCF x LCM = product. Do not copy that identity directly to three numbers.
Exit ticket
(1) Find HCF and LCM of 84 and 126 by prime factorisation. (2) Use the product relation to find a missing integer. (3) Decide whether a repeated-cycle situation requires HCF or LCM and justify the choice before calculating.