If a positive integer n, divided by 5 has a remainder 2

-----------------ASIDE----------------------------------
When it comes to remainders, we have a nice rule that says:

If N divided by D leaves remainder R, then the possible values of N are R, R+D, R+2D, R+3D,. . . etc.

For example, if k divided by 5 leaves a remainder of 1, then the possible values of k are: 1, 1+5, 1+(2)(5), 1+(3)(5), 1+(4)(5), . . . etc.
------ONTO THE QUESTION!!!------------------------

Positive integer n, divided by 5 has a remainder 2
Some possible values of n: 2, 7, 12, 17, 22, 27, 32, 37, . . . etc

Now let's examine the statements:
I. n is odd.
This need not be true.
Among the possible values of n, we see that n need not be odd
So statement 1 is FALSE

II. n+1 cannot be a prime number.
Not true.
Among the possible values of n, we see that n COULD equal 2
2+1 = 3, and 3 IS a prime number
So, n+1 CAN BE a prime number
So statement 2 is FALSE

NOTE: Since statements I and II are false, we need not examine statement III, since there are no answer choices that suggest that only statement III is true.
So, the correct must be A

RELATED VIDEO

If a positive integer n, divided by 5 has a remainder 2

Theory: Dividend = Divisor*Quotient + Remainder

n -> Dividend
5 -> Divisor
a -> Quotient (Assume)
2 -> Remainder

=> n = 5*a + 2 = 5a + 2

Now, lets take option choice and see if we can prove it wrong

I n is odd
n = 5a + 2
If we take a = 0. We get the value of n as
n = 0*2 + 2 = 0 + 2 = 2 => NOT odd => FALSE

II n + 1 cannot be a prime number
Same example as first one
We can have n = 2 => PRIME => FALSE

III (n + 2) divided by 7 has remainder 2
n = 5a + 2
=> n + 2 = 5a + 2 + 2 = 5a + 4
a = 0
n + 2= 5*0 + 4 = 4 => Remainder of n (=4) by 7 becomes 4 => FALSE

So, Answer will be A
Hope it helps!

Watch the following video to learn the Basics of Remainders

If a positive integer n, divided by 5 has a remainder 2, which of the following must be true

I. n is odd
II. n+1 cannot be a prime number
III. (n+2) divided by 7 has remainder 2

A. None
B. I only
C. I and II only
D. II and III only
E. I, II and III[/quote]

A positive integer n, divided by 5 has a remainder 2 --> \(n=5q+2\), so n could be 2, 7, 12, 17, 22, 27, ...

I. n is odd. Not necessarily true, since n could be 2, so even.

II. n+1 cannot be a prime number. Not necessarily true, since n could be 2, so n+1=3=prime.

III. (n+2) divided by 7 has remainder 2. Not necessarily true, since n could be 2, so n+2=4 and 4 divided by 7 has remainder 4.

Answer: A.

Hope it's clear.

Possible values of n are { 7 , 12 , 17 , 22 , 27 ................... }

Now, check the options -

I. n can be Odd/Even
II. n can be Prime / Non Prime
III. n can/can not have remainder 2

Thus, the answer will be (A)

We can express n as:

n = 5q + 2

Let’s now analyze each Roman numeral:

I. n is odd

If q = 2, then 5q + 2 = 12, so n does not have to be odd.

II. n+1 cannot be a prime number

If q = 2, then 5q + 2 = 12, so n + 1 = 13, which is prime. So II does not have to be true.

III. (n+2) divided by 7 has remainder 2

n + 2 = 5q + 4

If q = 2, then 5q + 4 = 14, which has a remainder of zero when divided by 7. So III does not have to be true.

Answer: A

When it comes to remainders, we have a nice property that says:
If N divided by D leaves remainder R, then the possible values of N are R, R+D, R+2D, R+3D,. . . etc.
For example, if k divided by 5 leaves a remainder of 1, then the possible values of k are: 1, 1+5, 1+(2)(5), 1+(3)(5), 1+(4)(5), . . . etc.

Given: When positive integer n is divided by 5, we get remainder 2
So, the possible values of n are: 2, 7, 12, 17, 22, 27, 32, . . . .

Now let's examine each statement....
I. n is odd
If n = 2, then n is NOT odd.
So, statement I need not be true.
Check the answer choices.... eliminate B, C and E, since they state that statement I is true.

II. n + 1 cannot be a prime number
If n = 2, then n+1 = 3, and 3 is prime.
Check the remaining answer choices.... eliminate D, since it states that statement I Iis true.

By the process of elimination, the correct answer is A.

ASIDE: Notice that we are able to arrive at the correct answer without having to analyze statement III.
We were able to do this because we eliminated incorrect answer choices after analyzing each statement.

Cheers,
Brent

RELATED VIDEO

N can take on values such as 2, 7, 12, 17, 22, etc.

Looking at the answer choices, we see that none of I, II, or III must be true.

Answer: A