Question

The product of three consecutive positive integers is divisible
(a) 4
(b) 6
(c) no common factor
(d) only 1​

Answer 1

Answer:

(d) only 1.

I hope u will like it

Answer 2

Answer:

(b) 6

Step-by-step explanation:

Let the numbers be n, (n + 1), (n + 2).

To find a proof for our answer we must use the Euclids Division Lemma, or we just need to know that,

Dividend = Divisor × Quotient + Remainder

Now, we must know that,

When 3 consecutive terms are multiplied, it can always be divisible 6, always by 1, and by 4 (75% of the times).

The option 'no common factor' is wrong.

Now,

I will show you how 6 is always able to divide the product of 3 consecutive positive terms.

You can also use this method with 4 to discover that, 4 divides only 3 of the 4 times.

Now,

6 = 2 × 3

And we know that,

According to the divisibility rule of 6, if a number is divisible by 2 and 3, then it is also divisible by 6.

So, basically we just have to prove that, the product of the 3 consecutive terms are divisible by 2 and 3.

Now, according to the Euclids Division Lemma,

When a number is divided by 2, its remainder will be 0 or 1, for some positive integer 'q'.

So,

n = 2q or 2q + 1

1st Case

n = 2q

In this case,

n = 2q is divisible by 2.

2nd Case

n = 2q + 1

Here,

n is not divisible,

The next consecutive term will be

n + 1 = (2q + 1) + 1

n + 1 = 2q + 2

n + 1 = 2(q + 1)

So,

In the second case,

n + 1 is divisible by 2.

Hence,

In any form, Product of 3 consecutive terms is always divisible by 2.

Now,

According to the Euclids Division Lemma,

When a number is divided by 3, its remainders will be 0, 1, or 2, for some positive integer 'q'.

So,

n = 3q or 3q + 1 or 3q + 2

1st Case

n = 3q

In this case n is divisible by 3.

2nd Case

n = 3q + 1

n is not divisible by 3 but,

Next term = n + 1

n + 1 = (3q + 1) + 1

n + 1 = 3q + 2

So,

(n + 1) is also not divisible by 3

Next term,

n + 2 = (3q + 1) + 2

n + 2 = 3q + 3

n + 2 = 3(q + 1)

Hence,

(n + 2) is divisible by 3.

3rd Case

n = 3q + 2

So,

n is not divisible by 3,

Next term,

n + 1 = (3q + 2) + 1

n + 1 = 3q + 3

n + 1 = 3(q + 1)

Hence,

n + 1 is divisible by 3.

So,

In any form, Product of 3 consecutive terms will be always divisible by 3.

Thus,

We proved that, always their product will be divisible by 2 and 3.

Hence,

When the product of the 3 consecutive positive terms are taken, it will always be divisible by 6.

Remember, since we are multiplying, even if one of the numbers are divisible, then the whole product will be divisible, but this is not valid for addition, OK!!

Now,

If you don't believe me, take any 3 consecutive terms on random, and take their products and then divide it by 6, you will always get a non-decimal answer in the calculator.

Use this same above proof method, to see that 4 divides the Product only 75% of the time.

Hope it helped and believing you understood it........All the best