Question

Five consecutive positive integers are chosen at random. If the average of the five integers is odd, what is the remainder when the largest of the five integers is divided by 4?

Answer 1

Answer:

The remainder would be 1 or 3

Step-by-step explanation:

Consider the 5 consecutive positive integers are,

x,  x + 1,  x + 2,  x + 3,  x + 4,

$Average =\frac{\text{Sum of all observation}}{\text{Number of observations}}$

Since, the average of these 5 numbers = $\frac{x+x+1+x+2+x+3+x+4}{5}$

$=\frac{5x+10}{5}$

= x + 2

If x + 2 = odd,

⇒ x = odd - 2 = odd - even = odd

⇒ x + 4 = odd + even = odd

∵ an odd number is represented by '2n + 1'

Where, n = 0, 1, 2, 3, ........

Now, 2n + 1 = 1( mod 4) if n = even

While, 2n + 1 = 3( mod 4) if n = odd,

Hence, when the largest of the five integers is divided by 4 remainder would be 1 or 3.