happy1992 wrote:
If a and b are positive integers, what is the remainder when 9^(2a+1+b) is divided by 10?
(1) a = 3
(2) b is odd.
Target question: What is the remainder when 9^(2a+1+b) is divided by 10?This is a great candidate for
rephrasing the target question. First recognize that this is a clever way of asking, "
What is the units digit of 9^(2a+1+b)?"
Notice that 153 divided by 10 equals 15 with remainder 3
Likewise, 3218 divided by 10 equals 321 with remainder 8
And 97 divided by 10 equals 9 with remainder 7
So, we can write....
REPHRASED target question: What is the units digit of 9^(2a+1+b)?IMPORTANT: We can RE-rephrase this target question in a way that makes it super easy to analyze the statements.
To see how, let's examine some powers of 9
9^1 =
99^2 = 8
19^3 = 72
99^4 = 656
1.
.
.
Notice that, when the exponent is ODD, the units digit is
9When the exponent is EVEN, the units digit is
1So, all we need to do is determine whether or not the exponent, (2a+1+b), is ODD or EVEN
To make things easier, we should recognize that 2a is EVEN for all integer values of a.
This means 2a+1 is ODD for all integer values of a.
So, if b is ODD, then 2a+1+b = ODD + ODD = EVEN, which means the units digit of 9^(2a+1+b) is
1And, if b is EVEN, then 2a+1+b = ODD + EVEN = ODD, which means the units digit of 9^(2a+1+b) is
9So, to answer the target question, all we need to know is whether b is odd or even
So,.......
RE-REPHRASED target question: Is n even or odd? Statement 1: a = 3 This is not enough information to determine whether
n is even or oddSince we cannot answer the
RE-REPHRASED target question with certainty, statement 1 is NOT SUFFICIENT
Statement 2: b is oddPerfect!!
Since we can answer the
RE-REPHRASED target question with certainty, statement 2 is SUFFICIENT
Answer: B
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