Question

Prove that the product of any
two three consecu
positive integers
is divisible by b.
The lcm of two number is 14 times their
Sum of Lom and HOF is bere
one number is 280. then find the other number ​

Answer 1

Answer:

It's equivalent to saying why is the product of 3 consecutive integers even. Well there are only 2 configurations possible for chosing such integers.

1. Odd-Even-Odd

2. Even-Odd-Even

So, you are bound to find at least one even number in the product.

Also note that the product of every 3 consecutive integers is also divisible by 3 !

You can try to prove the more general statement, by the Pigeonhole Principle that the product of every k consecutive integers, is divisible by 2,3,..k for every positive integer k>1.

The fun doesn't stop here - you can prove all kinds of crazy stuff. Note that for a given k , pick any 2 numbers n,m from 2,3,..k such that gcd(n,m)=1. Then the product of k consecutive integers is also divisible by nm !!

Have fun exploring more possibilities !