Re: If b and c are two-digit positive integers and b − c = 22d, is d
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20 Jun 2018, 09:16
Official Solution:-
If b and c are two-digit positive integers and b − c = 22d, is d an integer?
Step 1: Analyze the Question Stem
This is a Yes/No question. The question stem tells us that the variables b and c are integers between 10 and 99 inclusive and asks whether b − c is a multiple of 22.
Step 2: Evaluate the Statements:-
Statement (1) merely tells you that the two digits of b are the same, so the value of b must be 11, 22, 33, etc. Since you are
given no further information about c, Statement (1) by itself is insufficient, and answer choices (A) and (D) can be
eliminated.
Statement (2) provides the information that b + c is a multiple of 22, since b + c = 22x and x is an integer. However, the question asks about b − c, not b + c, so this statement, too, is insufficient. If you were unsure, you could plug in some number pairs to verify that b − c could be a multiple of 22 but need not be. Set b = 12 and c = 10; b + c = 22, which is divisible by 22, but b − c = 2, which clearly is not divisible by 22. Now try b = 11 and c = 33; b + c = 44 is divisible by 22, while b − c = −22 is divisible by 22 as well. Eliminate answer choice (B).
Now look at both statements together along with the information in the question stem. Statement (1) says that the tens and units digits of b are the same, so b is a multiple of 11. From statement (2), you know that b + c is a multiple of 22. Every multiple of 22 is a multiple of 11, so b + c is a multiple of 11. Since b + c is a multiple of 11 and b is a multiple of 11, (b + c) − b = c, which is a multiple of 11. Since b and c are both multiples of 11, b − c is also a multiple of 11. Since b + c is a multiple of 22, b + c is even. This means that both b and c are even or both b and c are odd. In either case, b − c is even, or, in other words, b − c is a multiple of 2. Since b − c is a multiple of 11, b − c is a multiple of 2, and the integers 2 and 11 have no common factor greater than 1, b − c is a multiple of 11 × 2 = 22. The statements taken together are sufficient to answer the question definitively yes.
Choice (C) is correct.