Discover that the HCF times the LCM of two numbers is their product, see why, and use it to find a missing value.
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The product of two numbers is always a common multiple.
Which is greater: the LCM of two numbers, or their product?
The book's hint: is the product also a common multiple of the two numbers? Yes: is a multiple of and of . So the lowest common multiple can never be bigger than the product.
Checking three pairs and spotting the pattern.
Problem. Is the LCM a factor of the product? If so, what must it be multiplied by? (Textbook p. 62)
HCF × LCM = product, for two numbers only.
Give the HCF and LCM as prime factorisations.
Figure it Out, Textbook p. 63, Q3. Write each answer as a product of primes, such as 2 × 3 × 5.
(a) 3 × 3 × 5 × 7 × 7 and 12 × 7 × 11 (write 12 as 2 × 2 × 3).
HCF =
LCM =
(b) 45 = 3 × 3 × 5 and 36 = 2 × 2 × 3 × 3.
HCF =
LCM =
Check (b): 9 × 180 = 1620 = 45 × 36.
Questions on HCF × LCM, co-prime pairs and the LCM of primes.
Two numbers have HCF 1 and LCM 66 (Textbook p. 63, Q4). Which pair works?
Explain HCF × LCM = product, and test it on three numbers.
Try This, Textbook p. 63.
Why does HCF × LCM = product? Explain with prime factorisations, using any pair you like. Hint from the book: some primes are common to both factorisations and the rest occur in only one. See where each kind goes in the HCF, the LCM and the product.
Does the property hold for 3 numbers? Test it, for example with 2, 3 and 4.
Pick two numbers that share a prime.
Shared primes, then the others.
Find HCF, LCM and product.