If n is a positive integer, what is the value of n?

Question

(1) For any positive integer y, the product of all integers from y to y+n is divisible by 8.
(2) n^2 - 5n + 6 = 0

gmatophobia · Answer

Let's start with statement 2, as that statement is simpler of the two statements.

Statement 2

n^2 - 5n + 6 = 0

(n-2) (n-3) = 0

So n can be either 2 or it can be 3

As we are getting two values of n, we can rule out. B and D.

Statement 1

For  any   positive integer y, the  product of all integers from y to y+n  is divisible by 8.

Let's see when can this happen.

If  y is odd , we need a minimum of four integers, including y, to be sure that the product will be divisible by 8.

y = 1, the product of all integers from y to y+n will be divisible by 8, when we have at least 3 twos (i.e.  2^3 ) in the product.

So 1 * 2 * 3 * 4

Let's take another value

y = 3; the minimum value of n should be 3.

Ex. 3 * 4 * 5 * 6

Now if  y is even , the minimum value of n should be 2.

Note, this is the minimum required value, any value greater than the minimum value also satisfies.

As n can take any value  \geq{3}   we can have multiple possible values. Hence we can rule out A.

Combined

Statement 2 limits the value of n to either 2 or to 3. Now as the nature of y is not known, we need to take the maximum of 2 and 3 to be sure and for Statement 1 to always hold true.

Hence the value of n = 3

The statements combined can answer the question.

Option C

YuktaSurana · Answer

Statement 1

In this case why n can't be 2 ?
 if n =2 and y =7
Hence, the product of all integers from y to y+n => 7*8 is divisible by 8

So value of n can be 2, 3 or 4 which makes (E) as the answer??

ajaychiratla · Answer

Aabhash777

How is the answer C, if n = 2 also works and we have no idea about Y, say Y = 2 and n= 2, then (2*3*4), this is also divisible by 8.

So answer should be E right?

ajaychiratla · Answer

Aabhash777

How is the answer C, if n = 2 also works and we have no idea about Y, say Y = 2 and n= 2, then (2*3*4), this is also divisible by 8.

So answer should be E right?

Flower_Child · Answer

Statement (1)  - For any positive integer y, the product of all integers from y to y+n is divisible by 8.

Notice that the question says product of ALL integers from y to y+n .
That means it is product of consecutive numbers.

We know that if we have two consecutive numbers, one will be even and other will be odd. 
If we have four consecutive numbers, two will be even and two will be odd. 
Out of the two even numbers , one will be surely divisible by 4 and other by 2, i.e their product will be divisible by 8.

So the minimum number of consecutive integers we need is 4 .

So if we have an integer y we need 3 more consecutive integers . So n=3 .
Now , this is the minimum number of consecutive integers that we can add . Even if we add 5 integers it will now be divisible by 8 as our required condition is already satisfied.
So we can't get the exact value of n . It could be anything greater than or equal to 3.

So insufficient .

Statement (2)  : n^2 - 5n + 6 = 0

(n-2) (n-3) = 0

So n can be either 2 or 3 .
 
Insufficient .

From statements (1) and (2)- 
We get the answer as 3.

So  C

MeherAbhijeet · Answer

Can anyone explain how n=2 ?
Since both n=2, 3 satisfy the equation.

ankitgoswami · Answer

Statement 1

y * (y+1) * .... * (y+n)

Now if it is divisble by 8 that means n>=3 as it will make atleast 2 terms even (such as 2 and 4) to fulfill the condition of divisibility by 8.

Insufficient

Statement 2

n^2-5n+6 = 0
n = 2 or 3

Insufficient

Combined

If n=2
Terms will be like 
y * (y+1) * (y+2) 
'A' says for any positive integer y, so let's take y=1
1*2*3 is not divisible by 8. Wrong

If n=3
Terms will be like 
y * (y+1) * (y+2) * (y+3) 
let's take y=1
1*2*3*4 is divisible by 8. It will also fulfill the condition hereafter. Correct

So we get n=3

Sufficient

Ans  C

Hope it helps.

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If n is a positive integer, what is the value of n?

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