LCM from the primes

The idea

To build the LCM, take every prime that occurs in any of the factorisations the maximum number of times it occurs in any one of them; this works for two or more numbers.

How many 2s?

96 = 2 × 2 × 2 × 2 × 2 × 3 and 360 = 2 × 2 × 2 × 3 × 3 × 5. A common multiple must contain both factorisations.

96 has five 2s. 360 has three 2s. At least how many 2s must the LCM have?

• 3
• 5
• 8

Build the LCM

For each prime, set how many beads go in the answer. A red dashed ring shows a bead that is left out. The toy says what is wrong.

The larger count, every time

The LCM must contain 96 and 360. For each prime it needs as many copies as the number that has more of it.

So the LCM of 96 and 360 is 2 × 2 × 2 × 2 × 2 × 3 × 3 × 5 = 1440. More 2s would give a common multiple too, but not the lowest. Take any bead away and one of the numbers is no longer inside.

To build the LCM, take every prime that occurs in any of the factorisations the maximum number of times it occurs in any one of them. This works for two or more numbers.

• Prime | Times in 96 and 360 | LCM takes
• 2 | 5 and 3 | 5
• 3 | 1 and 2 | 2
• 5 | 0 and 1 | 1

Notes

To build the LCM, take every prime that occurs in any of the factorisations the maximum number of times it occurs in any one of them. It works for two or more numbers.

Check yourself

Find the LCM of 12 = 2 × 2 × 3 and 18 = 2 × 3 × 3.

Answer: 36

2 appears twice in 12 and once in 18: take two. 3 appears once in 12 and twice in 18: take two. LCM = 2 × 2 × 3 × 3 = 36.

Which is the LCM of 12 = 2 × 2 × 3 and 30 = 2 × 3 × 5?

Find the LCM of three numbers: 4, 6 and 10.

Answer: 60

2: at most two (in 4). 3: one (in 6). 5: one (in 10). LCM = 2 × 2 × 3 × 5 = 60.

Dev says the LCM of 6 = 2 × 3 and 8 = 2 × 2 × 2 is 2 × 3 × 2 × 2 × 2 = 48. What is wrong?

• 2 × 3 = 6. That is the HCF: the smaller counts. The LCM needs every prime of both numbers.
• 2 × 2 × 3 × 2 × 3 × 5 = 360. That puts all the beads of both numbers together (12 × 30 = 360). It is a common multiple, but the shared 2 and 3 are counted twice. It is not the lowest.
• 2 × 2 × 3 × 5 = 60 — correct. Yes! Two 2s (the larger count), one 3 and one 5.
• 48 is not a multiple of 8.. 48 = 8 × 6, so it is a multiple of 8.
• 48 is a common multiple, but not the lowest: only three 2s are needed, not four. — correct. Yes! The most 2s in either number is three (in 8). The 2 in 6 can share a 2 with 8: the LCM is 2 × 2 × 2 × 3 = 24.
• Nothing, 48 is the LCM.. 24 is a common multiple of 6 and 8, and it is lower than 48.