Three Consecutive Natural NumbersDate: 09/17/2001 at 01:03:37 From: Jordan Harvey Subject: Three Consecutive Natural Numbers Why is it that the product of any three consecutive natural numbers is always divisible by six? Date: 09/17/2001 at 10:01:43 From: Doctor Sarah Subject: Re: Three Consecutive Natural Numbers Hi Jordan - thanks for writing to Dr. Math. Let's start with what we know about the divisibility rule for 6: If a number is divisible by both 2 and 3, it is also divisible by 6. (See "Divisibility Rules," from the Dr. Math FAQ: http://mathforum.org/dr.math/faq/faq.divisibility.html .) Now let's look at the natural numbers: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 ... - Every other number (2 4 6 8...) is even, and is therefore divisible by 2. - Every third number (3 6 9 12...) is a multiple of 3, and is therefore divisible by 3. So every combination of three consecutive natural numbers will include one even number and one multiple of three, and the product will therefore be divisible by 2 and 3, and so also by 6. - Doctor Sarah, The Math Forum http://mathforum.org/dr.math/ |
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