Consider the following statement about the positive integers a, b and n:
(×): ab is divisible by n
The condition 'either a or b is divisible by n' is:
- Anecessary but not sufficient for (×)
- Bsufficient but not necessary for (×)
- Cnecessary and sufficient for (×)
- Dnot necessary and not sufficient for (×)
Show the answer and worked solution
answer · B
- Anecessary but not sufficient for (×)
- Bsufficient but not necessary for (×)
- Cnecessary and sufficient for (×)
- Dnot necessary and not sufficient for (×)
Sufficiency is immediate: if n divides a, then a = kn and ab = knb, so n divides ab; the same works if n divides b. It is not necessary, because n can be built out of factors split between a and b. Take a = 2, b = 3, n = 6: then ab = 6 is divisible by 6, yet neither 2 nor 3 is. (The condition would be necessary if n were prime, which is exactly what the question does not say.)