Divisible by two numbers
The numbers in both lists are the multiples of the LCM.
Both 4 and 6
12 is divisible by 4 and by 6. So are 24 and 36.
A number is divisible by both 4 and 6. Which number must also divide it?
What this lesson covers
The idea
If A is divisible by k and by m, then A is divisible by LCM(k, m), since A's prime factorisation contains that of LCM(k, m); but a number divisible by k need not be divisible by every multiple of k.
Both 4 and 6
12 is divisible by 4 and by 6. So are 24 and 36.
A number is divisible by both 4 and 6. Which number must also divide it?
- 24 (4 × 6)
- 12
- 10 (4 + 6)
Two lists of multiples
Pick k and m. Multiples of k are blue, multiples of m are orange. The purple numbers are in both lists. Tap a number to see its prime beads.
Divisible by both: divisible by the LCM
If A is divisible by k and by m, then A is divisible by LCM(k, m), since A's prime factorisation contains that of LCM(k, m); but a number divisible by k need not be divisible by every multiple of k.
36 = 2 × 2 × 3 × 3 holds the beads of 12 = 2 × 2 × 3 and of 18 = 2 × 3 × 3. The LCM of 12 and 18 needs the beads 2 × 2 × 3 × 3 = 36, so every number divisible by both 12 and 18 is a multiple of 36.
42 is divisible by 7, but not by 28, which is also a multiple of 7.
Notes
If A is divisible by k and by m, then A is divisible by LCM(k, m), since A's prime factorisation contains that of LCM(k, m); but a number divisible by k need not be divisible by every multiple of k.
Check yourself
A number is divisible by 6 and by 15. It must be divisible by which number? (Give the LCM.)
Answer: 30
6 = 2 × 3 and 15 = 3 × 5. The LCM needs 2, 3 and 5: 2 × 3 × 5 = 30.
A number is divisible by both 4 and 10. Which statement is always true?
A number is divisible by 7. Is it divisible by 14?
How many numbers from 1 to 100 are divisible by both 6 and 8?
Answer: 4
LCM(6, 8) = 24. The multiples of 24 up to 100 are 24, 48, 72, 96. That is 4 numbers.
- It is a multiple of 8. 20 is divisible by 4 and by 10, but 20 is not a multiple of 8.
- It is a multiple of 14. 20 is divisible by 4 and by 10, but 20 is not a multiple of 14.
- It is a multiple of 20 — correct. Yes! LCM(4, 10) = 20, and every number divisible by both is a multiple of the LCM.
- It is a multiple of 40. 20 is divisible by 4 and by 10, but 20 is not a multiple of 40. 40 = 4 × 10 is bigger than the LCM.
- Always. 35 is divisible by 7 but not by 14.
- Never. 42 is divisible by 7 and by 14 (42 = 14 × 3).
- Sometimes: 42 is, but 35 is not — correct. Yes! 14 is a multiple of 7, but being divisible by 7 does not make a number divisible by every multiple of 7.