Math and Arithmetic

Are all numbers divisible by 3 and 4 divisible by 6?

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2011-01-26 06:57:13
2011-01-26 06:57:13

Yes. If a number is evenly divisible by both 3 and 4, it will also be divisible by 6.

This is because the prime factor of 3 is [3] -- in other words, 3 is a Prime number -- and the prime factors of 4 are [2, 2].

Thus, any number that is divisible by 3 and 4 will have as part of its prime factors the set [2, 2, 3]. (The smallest such number is 12, which is 2 x 2 x 3.) Since the prime factors of 6 are [2, 3] -- and [2, 3] is a subset of [2, 2, 3] -- then any such number must also be divisible by 6.

Another way to look at it is to note that all numbers which are divisible by both 3 and 4 are also divisible by 12 (which is 3 x 4). Since 12 is divisible by 6, then all multiples of 12 will also be divisible by 6.

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