Question

show that one and only one out of n, n+1, n+2, n+3, n+4 is divisible by 5

Answer 1

Step-by-step explanation:

Show that one and only one out of n, n+1, n+2, n+3, n+4 is divisible by 5

Let's put n in the format of:

• n=5q+r ,

where q is the quotient and r is the remainder

, 0≤r≤5

Depending on value of r, the format of n may become as: 5q, 5q+1, 5q+2, 5q+3, 5q+4, 5q+5

When n=5q and 5q+5, it is obvious that n is divisible by 5

Let's analyze the other values of n:

• Here we will put yes if number is divisible by 5, and no if not divisible

1. n=5q+1  

• n=5q+1 - no
• n+1 = 5q+2 -no
• n+2 = 5q+3 - no
• n+3 = 5q+4 - no
• n+4 = 5q+5= 5(q+1)  - yes

2. n=5q+2

• n=5q+2 - no
• n+1 = 5q+3 -no
• n+2 = 5q+4 - no
• n+3 = 5q+5= 5(q+1) - yes
• n+4 = 5q+6=5(q+1)+1 - no

3. n=5q+3

• n=5q+3 - no
• n+1 = 5q+4 -no
• n+2 = 5q+5= 5(q+1) - yes
• n+3 = 5q+6= 5(q+1)+1 - no
• n+4 = 5q+7=5(q+1)+2 - no

4. n=5q+4

• n=5q+4 - no
• n+1 = 5q+5= 5(q+1) -yes
• n+2 = 5q+6= 5(q+1)+1 - no
• n+3 = 5q+7= 5(q+1)+2 - no
• n+4 = 5q+8=5(q+1)+3 - no

As per above, in every case we have only one yes confirming that only one out of five numbers is divisible by 5

Hope is helps