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Questions 5-6

5. If the sum of five consecutive positive integers is a, then the sum of the next five consecutive integers in terms of a is
(A) a + 1
(B) a + 5
(C) a + 25
(D) 2a
(E) 5a

6. If P represents the product of the first 15 positive integers, then P is NOT a multiple of
(A) 99
(B) 84
(C) 72
(D) 65
(E) 57

Explanations: Questions 5-6

easy maths 5. If the sum of the first five consecutive integers in a set is a, then each of the next five consecutive integers will be five greater than the corresponding value of each integer in the original set. Since there are five integers in the new set, its value is a + 5(5) = a + 25. (C) is correct. You can easily prove this to yourself by picking numbers.

6. If P is a product of the first 15 integers, then ail of its factors will be combinations of these integers. P will not be a multiple of any number that has any different prime factor than P. The prime factors of P will be all the prime factors less than 15, that is 2, 3, 5, 7, 11, and 13. If any of the other answer choices have prime factors greater than this, they will not be factors of P.

Look at the answer choices, trying to find one that has a prime factor greater than 15. Start with choice (E) in questions in which you have to check the answer choices one by one. 57 factors to 19 3 3. Since 19 is prime and greater than 15, choice (E) is your answer. Once you've found an answer, don't waste time verifying the other choices.

Question 8 below is a number properties question with a bit of exponents mixed in. This is not uncommon. In the next section, you'll see some roots and exponents questions with a touch of number properties.

Questions 7-8

mathematic puzzle

7. If F< 0, which of the following represents the largest value?

(A)

(B)

(C)

(D)

(E) F - 1

8. If

and

, what is the minimum value of xy2?
(A)

(B)

(C)

(D)

(E)

Explanations: Questions 7-8

roots and exponents

7. Notice that in each of the fractional answer choices, the coefficient of F in the numerator is the same as the denominator. The fraction in each answer choice can be broken down into 2 new fractions, so rewrite each answer choice and compare their values:

(A)

(B)

(C)

(D)

(E) F - 1

In each answer choice a number is subtracted from F. Since the value of F is the same in each answer choice, just compare the numbers. The smaller the number you're subtracting, the larger the value of the entire expression, so choice (B) has the largest value. If you were confident of your ability with negative numbers and fractions, you could have picked some simple negative number for F, plugged it into the answer choices, and seen which was the largest.

8. Since x is negative and у is positive (the square of any number, except zero, is positive), xy1 is negative, since a positive times a negative is negative. The minimum value of xy2 occurs when xy1 is as negative as possible, that is, farthest from zero. Take the values of x and у that have the largest absolute values:

and

Then

, making (D) correct

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