Question

the positive integer n= (n+4)(n+5)(n+6)(n+7) is not divisible by which of the following number

Answer 1

Consider the positive integer is of the form

5q,5q+1,5q+2,5q+3

Here

b=5

r=0,1,2,3,4,

Where r=0,thenn=5q

Now, n=5q is divisible by 5

n+4=5q+4[not divisible by 5]

n+8=5q+8[not divisible by 5]

n+6=5q+6[not divisible by 5]

n+12=5q+12[not divisible by 5]

Where r=1,n=5q+1

n=5q+1

n+4=5q+5[divisible by 5]

n+8=5q+9[not divisible by 5]

n+6=5q+7[not divisible by 5]

n+12=5q+13[not divisible by 5]

Where r=2,n=5q+2

n=5q+2

n+4=5q+6[not divisible by 5]

n+8=5q+10[divisible by 5]

n+6=5q+8[not divisible by 5]

n+12=5q+14[not divisible by 5]

Where r=3,n=5q+3

n=5q+3

n+4=5q+7[not divisible by 5]

n+8=5q+11[not divisible by 5]

n+6=5q+9[not divisible by 5]

n+12=5q+15[ divisible by 5]

When,

r=4,n=5q+4

n=5q+4

n+4=5q+8[not divisible by 5]

n+8=5q+12[not divisible by 5]

n+6=5q+10[ divisible by 5]

n+12=5q+16[not divisible by 5]

From 1, 2, 3, 4, 5 Its clear that, one and only one out of n,n+4,n+12,n+6 is divisible by 5

Hence, this is the answer.