a positive integer n is called special if n is divisible by 4, N+1 is divisible by 5 and n+2 is divisible by 6. how many special integers smaller than 1000 are there?

The digits n, n+1 and n+2 implies that the special numbers are consecutive i.e. they follow one another. The count of special numbers less than 1000 is 17.

Given that:

First, calculate the LCM of 4, 5 and 6.

So, the LCM is:

This means that the special digit occurs after every other 60 digits (i.e. the difference between a special digit and another is 60)

To calculate the count of special digits, we make use of arithmetic progression formula

Where

--- the last term

--- the first special digit

--- the common difference as explained above

So, we have:

Collect like terms

Divide both sides by 60

Add 1 to both sides

Rewrite as:

Remove the decimal part (do not approximate)

Hence, the special numbers are 17 in total (that are less than 1000).

Read more about arithmetic progression at:

to find the total number of combinations, multiply the quantity of each item by each other:

2 pairs of shoes x 3 pairs of shorts = 6 different combinations.