When the integer n is divided by 8, the remainder is 5. Which of the following is NOT an even number?

A. n + 3
B. n/2 + 3.5
C. n – 3
D. 3n + 1
E. 5n + 2

(E) The most efficient way to solve this problem is to substitute a possible value of n into the answer choices and see which one is not an even number. n could be 5, 13, 21, etc. Let’s substitute these trial numbers.

Let’s choose 5:
(A) 5 + 3 = 8. This is EVEN.
(B) 5/2 + 3.5 = 6. This is EVEN.
(C) 5 – 3 = 2. This is EVEN.
(D) 3(5) + 1 = 16. This is EVEN.
(E) 5(5) + 2 = 27. This is ODD.

The correct answer is choice (E), since this is the only answer choice that produced an odd with n = 5.

Note: 5 divided by 8 is 0 with a remainder of 5. When you are looking for the remainder, you should stop dividing when continuing to do so would result in a decimal. For example, 13 divided by 8 has a remainder of 5. Dividing until you have 1.625 is not helpful here.
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As per the explantion 5 is taken as the example. However when 5 is divided by 8 the remainder is not 5 as it said in the question.

So I took an example as 50. When 50 is divided by 8 the remainder is 5 and thus 50 + 3 = 53 odd so answer is A.
Please explain.

For positive integers n and b,
If n = m × b + k,
where k < b, m and k are non-negative integers,
then k is the remainder when n is divided by b.

Note, that any dividend which is less than the divisor is also a remainder, because n = 0 × b + n.

The practical way to find the remainder is to divide using log division method.

Let's consider the given question.
5 = 0 × 8 + 5
Therefore 5 is the remainder when 5 is divided by 8.

When 50 is divided by 8 the remainder is 2, since 50 = 6 × 8 + 2.

The answer choice E is odd.
Plug in n = 5 to get
A. 8
B. 6
C. 2
D. 16
E. 27


In general, the proof is the following.
n = m × 8 + 5 is given by the question statement. Plug it in E.
E. 5n + 2 = 5(8m + 5) + 2 = 40m + 27 = even + odd = odd

I like the plug in method.