GRE_Math_Bible_eBook

As defined, 2! = 2 1 = 2 and 8! = 8 7 6 5 4 3 2 1. Hence, in Column B, we have

2! 8! = 2(8 7 6 5 4 3 2 1)

Similarly, 1! equals the product of the integers from 1 through 1. Hence, 1! = 1. Also, 9! = 9 8 7 6 5 43 2 1. Hence, in Column A, we have

1! 9! = 1(9 8 7 6 5 4 3 2 1) = 9 8 7 6 5 4 3 2 1

Now, rewriting 9 8 7 6 5 4 3 2 1 as 9(8 7 6 5 4 3 2 1) yields

2.According to the definitions given, 12∩15 equals (12*15) $ (12$15) = (GCF of 12 and 15) $ (LCM of

12and 15) = 3$60 = LCM of 3 and 60 = 60. The answer is (E).

3.Prime factoring 48 yields 2 2 2 2 3. The even factors of 48 are

2

2 2 (= 4)

2 2 2 (= 8)

2 2 2 2 (= 16)

3 2 (= 6)

3 2 2 (= 12)

3 2 2 2 (= 24)

3 2 2 2 2 (= 48)

Not counting the last factor (48 itself), the total number of factors is 7. Hence, Column A equals 7.

Now, 122 = 2 61 (61 is a prime). So, 2 is the only even factor. Hence, *(122) = 1, which is less than 7 (= Column A). Hence, the answer is (A).

Hard

4. The only factors of a prime number are 1 and itself. Hence, π(any prime number) = 2. So, π(a) = 2 and π(b) = 2, and therefore Column A equals π(a) + π(b) = 2 + 2 = 4.

Now, the factors of ab are 1, a, b, and ab itself. Hence, the total number of factors of ab is 4. In other words, π(a × b) = 4. Hence, Column B also equals 4.

Since the columns are equal, the answer is (C).

5. By the given definition, ∆(n) = (n + 4)(n + 5)(n + 6), a product of three consecutive integers. There is exactly one multiple of 3 in every three consecutive positive integers. Also, at least one of the three numbers must be an even number. Hence, ∆(n) must be a multiple of both 2 and 3. Hence, ∆(n) must be a multiple of 6 (= 2 × 3), because 2 and 3 are primes. The answer is (C).

32GRE Math Bible

6. Working from the inner parentheses out, we get

((–π)*)* = (π/(–π))* =

(–1)* =

π/(–1) =

–π

The answer is (C).

Method II:

We can rewrite this problem using ordinary function notation. Replacing the odd symbol x* with f (x) gives f (x) = π / x . Now, the expression ((–π)*)* becomes the ordinary composite function

f(f(–π)) = f(π/(–π) =

f(–1) =

π/(–1) =

–π

7. The function a * b is defined to be a/b – b/a for any numbers a and b. Applying this definition to the columns gives

The given inequality m > n > 0 indicates that both m and n are positive and therefore their product mn is positive. Multiplying both columns by mn to clear fractions yields

Adding n2 and m2 to both columns yields

Finally, dividing both columns by 2 yields

Since both m and n are positive and m > n, m2 > n2. Hence, Column B is greater than Column A, and the answer is (B).

8. The numbers 102 and 729 are themselves multiples of 3. Also, a multiple of 3 exists once in every three consecutive integers. Counting the multiples of 3 starting with 1 for 102, 2 (= 1 + (105 – 102)/3 = 1 + 1 = 2) for 105, 3 (= 1 + (108 – 102)/3 = 1 + 2 = 3) for 108, and so on, the count we get for 729 equals 1 + (729

– 102)/3 = 1 + Column B. Hence, Column A is greater than Column B by 1 unit. Hence, the answer is (A).

34GRE Math Bible

8.Know these rules for radicals:

A.xy = xy

B.xy = xy

9.Pythagorean Theorem (For right triangles only):

(C)8

Since the triangle is a right triangle, the Pythagorean Theorem applies: h2 + 32 = 52 , where h is the height

1

of the triangle. Solving for h yields h = 4. Hence, the area of the triangle is 2 (base)(height)=

12 (3)(4) = 6. The answer is (A).

10. When parallel lines are cut by a transversal, three important angle relationships are formed:

11.In a triangle, an exterior angle is equal to the sum of its remote interior angles and therefore greater than either of them.

a

e = a + b and e > a and e > b

12. A central angle has by definition the same measure as its intercepted arc.

60˚

60˚

38GRE Math Bible

The following properties for odd and even numbers are very useful—you should memorize them:

even × even = even odd × odd = odd even × odd = even

even + even = even odd + odd = even even + odd = odd

Example 2: If n is a positive integer and (n + 1)(n + 3) is odd, then (n + 2)(n + 4) must be a multiple of which one of the following?

(n + 1)(n + 3) is odd only when both (n + 1) and (n + 3) are odd. This is possible only when n is even. Hence, n = 2m, where m is a positive integer. Then,

(n + 2)(n + 4) = (2m + 2)(2m + 4) = 2(m + 1)2(m + 2) = 4(m + 1)(m + 2) =

4 × (product of two consecutive positive integers, one which must be even) = 4 × (an even number), and this equals a number that is at least a multiple of 8

Hence, the answer is (D).

Consecutive integers are written as x, x + 1, x + 2, . . .

Consecutive even or odd integers are written as x, x + 2, x + 4, . . .

The integer zero is neither positive nor negative, but it is even: 0 = 2 × 0.

A prime number is an integer that is divisible only by itself and 1.

The prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, . . .

A number is divisible by 3 if the sum of its digits is divisible by 3.

For example, 135 is divisible by 3 because the sum of its digits (1 + 3 + 5 = 9) is divisible by 3.

A common multiple is a multiple of two or more integers.

For example, some common multiples of 2 and 5 are 0, 10, 20, 40, and 50.

The least common multiple (LCM) of two integers is the smallest positive integer that is a multiple of both.

For example, the LCM of 4 and 10 is 20. The standard method of calculating the LCM is to prime factor the numbers and then form a product by selecting each factor the greatest number of times it occurs. For 4 and 10, we get

4 = 22

10 = 2 • 5

In this case, select 22 instead of 2 because it has the greater number of factors of 2, and select 5 by default since there are no other factors of 5. Hence, the LCM is 22 • 5 = 4 • 5 = 20.

For another example, let’s find the LCM of 8, 36, and 54. Prime factoring yields

8 = 23

36 = 22 • 32

54 = 2 • 33

In this case, select 23 because it has more factors of 2 than 22 or 2 itself, and select 33 because is has more factors of 3 than 32 does. Hence, the LCM is 23 • 33 = 8 • 27 = 216.

A shortcut for finding the LCM is to just keep adding the largest number to itself until the other numbers divide into it evenly. For 4 and 10, we would add 10 to itself: 10 + 10 = 20. Since 4 divides evenly in to 20, the LCM is 20. For 8, 36, and 54, we would add 54 to itself: 54 + 54 + 54 + 54 = 216. Since both 8 and 36 divide evenly into 216, the LCM is 216.

The absolute value of a number, | |, is always positive. In other words, the absolute value symbol eliminates negative signs.

For example, −7 = 7 and −π = π . Caution, the absolute value symbol acts only on what is inside the symbol, . For example, − − (7− π) = −(7− π). Here, only the negative sign inside the absolute value symbol but outside the parentheses is eliminated.

Example 3: The number of prime numbers divisible by 2 plus the number of prime numbers divisible by 3 is

A prime number is divisible by no other numbers, but itself and 1. Hence, the only prime number divisible by 2 is 2 itself; and the only prime number divisible by 3 is 3 itself. Hence, The number of prime numbers divisible by 2 is one, and the number of prime numbers divisible by 3 is one. The sum of the two is 1 + 1 = 2, and the answer is (C).

Solving the given equation 15x + 16 = 0 yields x = –16/15.

Substituting this into the expression 15 x yields

The answer is (D).

The product (quotient) of positive numbers is positive.

The product (quotient) of a positive number and a negative number is negative.

For example, –5(3) = –15 and −63 = −2.

The product (quotient) of an even number of negative numbers is positive.

For example, (–5)(–3)(–2)(–1) = 30 is positive because there is an even number, 4, of positives.

−9 9

−2 = 2 is positive because there is an even number, 2, of positives.

The product (quotient) of an odd number of negative numbers is negative.

For example, (−2)(−π)(−3) = −2π 3 is negative because there is an odd number, 3, of negatives.

The sum of negative numbers is negative.

For example, –3 – 5 = –8. Some students have trouble recognizing this structure as a sum because there is no plus symbol, +. But recall that subtraction is defined as negative addition. So –3 – 5 = –3 + (–5).

A number raised to an even exponent is greater than or equal to zero.

For example, (−π )4 = π 4 ≥ 0 , and x2 ≥ 0 , and 02 = 0 0 = 0 ≥ 0 .