Question

"The product of 3 consecutive integers is divisible by 6. " Is this statement true? Give reason for your answer.

Class 10 Mathematics. Chapter - Real Numbers. Answer with explanation.

Answer 1

Answer:

Yes, the product of any three consecutive integers is divisible by 6.

Step-by-step explanation: Let the three consecutive integers be a, a+1 and a+2.

We know, a number is divisible by 6, if it is divisible by both 2 and 3.

Hence, let us check the divisibility of a,a+1 and a+2 by 2 and 3.

Divisibility by 2: Whenever any number is divided by 2, the only possible remainders are 0 and 1.

Thus, if i is any integer, then,

a = 2i, 2i +1 or 2i+2.

• If a = 2i, then the number is surely divisible by 2.
• If a = 2i + 1, then a+1 = 2i + 1 + 1.

                                 ⇒ a+1 = 2i +2

                                 ⇒ a+1 = 2 (i+1)

hence, a+1 is divisible by 2.

• If a = 2i+2, then a = 2(i+1) which is again divisible by 2.

Thus, we see that, one of the numbers among a, a+1, a+2 is always divisible by 2.

Divisibility by 3: Whenever any number is divided by 3, the only possible remainders are 0, 1 and 2.

Thus, if i is any integer, then,

a = 3i, 3i +1 or 3i+2.

• If a = 3i, then the number is surely divisible by 3.
• If a = 3i + 1, then a+2 = 3i + 1 + 2.

                                 ⇒ a+2 = 3i +3

                                 ⇒ a+2 = 3 (i+1)

hence, a+2 is divisible by 3.

• If a = 3i+2, then a + 1 = 3i+2+1

                               ⇒ a+1 = 3i +3

                               ⇒ a+1 = 3 (i+1)

which is again divisible by 3.

Thus, we see that, one of the numbers among a, a+1, a+2 is always divisible by 3.

Since we established the divisibility by both 2 and 3, we can safely conclude that the product of 3 consecutive numbers is always divisible by 6.