From the answers we can see that the list of factors will be relatively small, so it s easiest just

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to list them out The pairs of factors are 1 and 342, 2 and 171, 3 and 114, 6 and 57, 9 and 38, and 18 and 19 That makes 12 factors

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9 B

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Once we simplify 27p 32 = 34 98, we get (33)p 32 = 34 (32)8, or 33p 32 = 34 316 That is the

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same as 3p + 2 = 4 + 16, so p must equal 6

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x We can take this question directly into an equation, which would look like 200 = 40 100 When we solve, we get 500, choice E The prime factors less than 10 are 2, 3, 5, and 7, so the product of any two of those cannot be

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correct That eliminates choice A (the product of 2 and 3), C (the product of 2 and 5), D (the product of 2 and 7), and E (the product of 3 and 5) Since 9 is the product of 3 and 3, which are not distinct (or 1 and 9, which are not prime), B is the correct answer

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x This question can also be put directly into equation form: 180 = 45 When we solve, we 100 get 025, choice C Simplifying gives us (05)3, which is 0125 The positive factor pairs for 180 are: 1 and 180, 2 and 90, 3 and 60, 4 and 45, 5 and 36, 6 and

13 A 14 A

30, 9 and 20, 10 and 18, and 12 and 15 The positive factor pairs for 96 are: 1 and 96, 2 and 48, 3 and 32, 4 and 24, 6 and 16, and 8 and 12 They have 1, 2, 3, 4, 6, and 12 in common for a total of 6 factors

15 C

Approximating the equation gives us

30 2 1 , which simplifies to 120 2

BASIC PRINCIPLES OF NUMBERS

16 D

Of the answer choices, only 9 is not a perfect cube, which means that its cube root is a non-

integer

17 D

Statement (1) tells us that r = 25 or 25, but when you look at the question stem, the radical sign for r is present, which only allows for a positive root So r = 5 Therefore statement (1) is

sufficient and we can eliminate B, C, and E Statement (2) also gives us enough information to solve for r If we rearrange the equation in the t t t question stem, we can get on one side, giving us r = Since = 5, r = 5 So statement (2) s s s is also sufficient, and our answer is D

18 C

Statement (1) does not give us a definite yes or no answer, because b could be a lot of different

numbers So we can eliminate A and D Statement (2) is also not sufficient, as a and b could equal 1 and 2, or 05 and 2, or 2 and 4, or an infinite number of other variations in which a is smaller than b So B is not correct Putting them together, we know that a = 2, and b must be larger than a to satisfy Statement (2) ab then must be larger than 2 and the answer to the question is definitely yes C is the correct answer

19 E

Statement (1) is not sufficient because f could be less than g or it could just be g + 05 We can

cross off A and D Statement (2) is not sufficient because while the absolute value of f must be less than the absolute value of g, the actual value of g could still be negative while f could be positive The statement doesn t give us enough information, so we can eliminate B Putting the statements together doesn t really give us any more information, so the correct answer is E

20 C

Because statement (1) just tells us that p could be any negative number, it is not sufficient to

answer the question A and D can be eliminated Statement (2) alone tells us that p could be either 3 or 3, which would give different values for 12 , so it is also not sufficient and we can eliminate B 5p The statements together tell us that p must be 3, so we can solve for answer 12 and C is the correct 5p