Numbers

vijaynarayanan wrote:If n is a positive integer, what is the remainder when 38n+3 + 2 is divided by 5?
A. 0
B. 1
C. 2
D. 3
E. 4

Correct Answer: E

dude please verify the question

sorry my mistake. I have corrected the question now.

It should read [3 to the power of (8n+3)] plus 2

3^(8n+3)+2=5q+r

3^(8n+3)+2=27.3^8n + 2=27.81^2n+2

Note watever the value of n,27.81 will have unit digit as 7. 7+2 ends in 9.

So when it's divisible by 7,the remainder is 4

Please correct me if I am wrong

We know that whatever positive integer we pick for n, we will arrive at the same result (otherwise they wouldn't have been able to ask the question this way).

So, let n be the smallest, easiest positive integer you can think of: 1.

So, we have 3^(8(1) + 3) or 3^11. When you have an exponent + remainder question, you should either recall or work out the cyclity of the units digit for that base (when you raise to consecutive-integer powers).:

3^1 = 3
3^2 = 9
3^3 = 27
3^4 = 81
3^5 = 243

Look at the units digits. They went from 3, 9, 7, 1, and then back to 3 again. So, this pattern of 3, 9, 7, 1 will continue. That is, the 1st, 5th, 9th, 13th powers of 3 will all have units digit of 3 (the 2nd, 6th, 14th powers will all have units digit of 9, and so on).

We want the units digit of 3^11. There are four numbers in the cycle. 11/4 = 2R3. "R3" tells us which number in the cycle our units digit is. So, the units digits of 3^11 is the 3rd number in the cycle, or 7. Once we add the "2", our units digit will be 9. 9/5 leaves a remainder of 4, or choice E.