# 1.Show that n(n+1)(n+2) is divisible by 6. 2.Show that #1^2015+2^2015+3^2015+4^2015+5^2015+6^2015# is divisible by 7. How do I solve these?

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I have no lead on the first problem. I know the divisibility rule of 7 but I have to know the whole number for that whereas I can only know the last few digits.

I have no lead on the first problem. I know the divisibility rule of 7 but I have to know the whole number for that whereas I can only know the last few digits.

##### 2 Answers

For the first problem, note that a number is divisible by

For the second problem, we can solve this using modular arithmetic. The basic idea behind modular arithmetic is that rather than look at the specific value of a given integer, we look at its remainder when divided by a given *modulus*. This is just like how when we use an analog clock, we will arrive at the same time if we wait *modulo* *congruent modulo 12* (and we use the symbol

As a number is divisible by

As

As

As

As

As

As

Then, substituting our newly found values into the given expression, we have

Therefore, as the expression is congruent to

Alternative way

#### Explanation:

For the **second part** of the problem this can be proved for first

We know that **when n is odd.** It can be easily verified by putting **a** when the value of **zero**

So if we rearrange the given problem as below

we see that here sum of the bases in every pair with in parentheses

like **7**. So as per the rule of divisibility the given sum is divisible by 7

similarly the following series of 12 terms is also divisible by **13**