Question

How many positive multiples of 10 that are less than 1000 are the sum of 4 consecutive integers?

Answer 1

Answer:

There are 50 numbers.

Explanation:

We can check,

1 + 2 + 3 + 4 = 10 (multiple of 10)

Now we need  to find how many multiples of 10 can be written in same manner.

Let the first of the four numbers be x. So ,

Four consecutive numbers are: x, x+1, x+2, x+3

x + (x+1) + (x+2) + (x+3) = 10n

Where n represent the integer

4x  + 6 = 10n

divide by 2 both the sides and solve for x

$x = \frac{5n-3}{2}$

We can also write this as,

x = $(2n-1) +\frac{n-1}{2}$

So x is integer for n = an odd number. So for n = 1, 3, 5, .... 99 we can write a number as a sum of four consecutive intezers.

So, there are 50 numbers.