Numbers

[vijaynarayanan]

If n is a positive integer, what is the remainder when 2 + 3^(8n+3) is divided by 5?
A. 0
B. 1
C. 2
D. 3
E. 4

Correct Answer: E

[amising6]

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

[vijaynarayanan]

sorry my mistake. I have corrected the question now.

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

[selango]

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

[Testluv]

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.