Question

he product of three consecutive integers is divisible by
(a) 2
(b) 3
(c) 5
(d) 6​

Answer 1

Answer:

(a) 2

Step-by-step explanation:

x. x+1.x+2=x3.2

x3.2is divisible b

Answer 2

Three consecutive integers  product will be divisible by 6

Let the three consecutive positive integers be = n, n + 1 and n + 2.  

When a number is divided by 3, the remainder obtained will be either = 0 or 1 or 2.  

Therefore, n = 3p or 3p + 1 or 3p + 2, where p is an integer.

Thus,  

If n = 3p, then n will be divisible by 3.  

If n = 3p + 1, → n + 2 = 3p + 1 + 2 = 3p + 3 = 3(p + 1) will be divisible by 3.  

If n = 3p + 2, → n + 1 = 3p + 2 + 1 = 3p + 3 = 3(p + 1) will be divisible by 3.  

Hence, one of the numbers from n, n + 1 and n + 2 is always divisible by 3.  

Similarly, when a number will be divided by 2, the remainder obtained will be 0 or 1.  

Therefore, n = 2q or 2q + 1, where q is an integer.  

If n = 2q = n and n + 2 → 2q + 2 = 2(q + 1) divisible by 2.  

If n = 2q + 1 → n + 1 = 2q + 1 + 1 = 2q + 2 = 2 (q + 1) divisible by 2.  Hence, one of the numbers among n, n + 1 and n + 2 will be  always divisible by 2.  

Hence, numbers n (n + 1) (n + 2) is divisible by both 2 and 3.

Therefore, the numbers n (n + 1) (n + 2) are divisible by 6.