If a and b are positive integers, what is the remainder when \(4^{2a+1+b}\) is divided by 10?

This is a classic "C trap" question: "C trap" is a problem which is VERY OBVIOUSLY sufficient if both statements are taken together. When you see such question you should be extremely cautious when choosing C for an answer.

Back to the question: 4 in positive integer power can have only 2 last digits: 4, when the power is odd or 6 when the power is even. Hence, to get the remainder of 4^x/10 we should know whether the power is odd or even: if it's odd the remainder will be 4 and if it's even the remainder will be 6.

(1) a = 1 --> \(4^{2a+1+b}=4^{3+b}\) depending on b the power can be even or odd. Not sufficient.

(2) b = 2 --> \(4^{2a+1+b}=4^{2a+3}=4^{even+odd}=4^{odd}\) --> the remainder upon division of \(4^{odd}\) by 10 is 4. Sufficient.

Answer: B.




The power of 4 is \(2a+3\) and since \(a\) is a positive integer then the lowest value of \(2a+3\) is 5, for \(a=1\). Next, even if the power were 1 then 4^1=4 and the remainder upon division of 4 by 10 would still be 4.

Hope it's clear.
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REM(4^(2a+1+b))/10

Means we have to find last digit of the expression.So rephrasing the question

What is the last digit of 4^(2a+1+b)

(1).
a=1

Break the expression as 4^2a * 4^1 * 4^b.

b is unknown hence INSUFFICIENT

(2).

b=2

4^2a * 4^1 * 4^b.

If you can observe the expression 4^2a, you will see that this will always give last digit as '6' you can try out numbers if you want.

So knowing the expression and value of b last digit can be calculated and hence the remainder can also be calculated.

Hence (B) it is !!
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We need to determine the remainder when 4^(2a+1+b) is divided by 10. Since the remainder when an integer is divided by 10 is always equal to the units digit of that integer, we need to determine the units digit of 4^(2a+1+b).

We can also simplify 4^(2a+1+b):

4^(2a+1+b) = (4^2a)(4^1)(4^b)

Let’s now evaluate the pattern of the units digits of 4^n for positive integer values of n. That is, let’s look at the pattern of the units digits of powers of 4. When writing out the pattern, notice that we are ONLY concerned with the units digit of 4 raised to each power.

4^1 = 4

4^2 = 6

4^3 = 4

4^4 = 6

The pattern of the units digit of powers of 4 repeats every 2 exponents. The pattern is 4–6. In this pattern, all exponents that are odd will produce a 4 as its units digit and all exponents that are even will produce a 6 as its units digit. Thus:

4^1 has a units digit of 4 and 4^2a has a units digit of 6.

Thus, to determine the units digit of (4^2a)(4^1)(4^b), we only need the value of b.

Statement One Alone:

a = 1

Since we do not have the value of b, statement one alone is not sufficient to answer the question.

Statement Two Alone:

b = 2

Since we have the value of b, statement two is sufficient to determine the remainder when (4^2a)(4^1)(4^b) is divided by 10.

Answer: B
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