SAT Math Hard Practice Quiz
Numbers and Operations 5. How many integers between 10 and 500 begin and end in
3?
1. A bag contains tomatoes that are either green or red.
The ratio of green tomatoes to red tomatoes in the bag
is 4 to 3. When five green tomatoes and five red tomatoes
are removed, the ratio becomes 3 to 2. How many red
tomatoes were originally in the bag?
(A) 12 6. A particular integer N is divisible by two different prime
(B) 15 numbers p and q. Which of the following must be true?
(C) 18
(D) 24 I. N is not a prime number.
(E) 30
II. N is divisible by pq.
III. N is an odd integer.
2. If each digit in an integer is greater than the digit to the (A) I only
left, the integer is said to be “monotonic”. For example, (B) II only
12 is a monotonic integer since 2 > 1. How many positive (C) I and II only
two-digit monotonic integers are there? (D) I and III only
(E) I, II, and III
(A) 28
(B) 32
(C) 36
(D) 40 7. A perfect square is an integer that is the square of an
(E) 44 integer. Suppose that m and n are positive integers such
that mn > 15. If 15mn is a perfect square, what is the
least possible value of mn ?
a, 2a − 1, 3a − 2, 4a − 3, . . .
3. For a particular number a, the first term in the sequence
above is equal to a, and each term thereafter is 7 greater
than the previous term. What is the value of the 16th
term in the sequence? 8. M is a set of six consecutive even integers. When the
least three integers of set M are summed, the result is x.
When the greatest three integers of set M are summed,
the result is y. Which of the following is true?
(A) y = x − 18
3
4. If p is a prime number, how many factors does p have? (B) y = x + 18
(C) y = 2x
(D) y = 2x + 4
(A) One (E) y = 2x + 6
(B) Two
(C) Three
(D) Four
(E) Five
erikthered.com/tutor pg. 1
, SAT Math Hard Practice Quiz
9. A three-digit number, XYZ , is formed of three different 10. An integer is subtracted from its square. The result could
non-zero digits X, Y , and Z. A new number is formed by be which of the following?
rearranging the same three digits. What is the greatest
possible difference between the two numbers? (For ex-
ample, 345 could be rearranged into 435, for a difference (A) A negative integer.
of 435 − 345 = 90.) (B) An odd integer.
(C) The product of two consecutive even integers.
(D) The product of two consecutive odd integers.
(E) The product of two consecutive integers.
erikthered.com/tutor pg. 2
, SAT Math Hard Practice Quiz
Algebra and Functions 4. Let m and n be positive integers such that one-third of
m is n less than one-half of m. Which of the following is
a possible value of m ?
1. Let m be √
an even integer. How many possible values of
m satisfy m + 7 ≤ 3 ?
(A) 15
(B) 21
(A) One (C) 24
(B) Two (D) 26
(C) Three (E) 28
(D) Four
(E) Five
5. If a and b are numbers such that (a − 4)(b + 6) = 0, then
x+3 what is the smallest possible value of a2 + b2 ?
2. Let x be defined by x = for any x such that
x−1
x 6= 1. Which of the following is equivalent to x − 1 ?
x+2
(A)
x−1
4 6. Let f (x) = ax2 and g(x) = bx4 for any value of x. If a
(B) and b are positive constants, for how many values of x is
x−1
f (x) = g(x) ?
2x + 4
(C)
x−1
2 (A) None
(D) (B) One
x−1
(C) Two
x+2 (D) Three
(E)
x−2 (E) Four
3 2
3. Let a and b be numbers such √ that a = b . Which of the 7. Let a and b be numbers such that 30 < a < 40 and 50 <
following is equivalent to b a ?
b < 70. Which of the following represents all possible
values of a − b ?
(A) b2/3
(A) −40 < a − b < −20
(B) b4/3 (B) −40 < a − b < −10
(C) −30 < a − b < −20
(C) b2 (D) −20 < a − b < −10
(E) −20 < a − b < 30
(D) b3
(E) b4
erikthered.com/tutor pg. 3
Numbers and Operations 5. How many integers between 10 and 500 begin and end in
3?
1. A bag contains tomatoes that are either green or red.
The ratio of green tomatoes to red tomatoes in the bag
is 4 to 3. When five green tomatoes and five red tomatoes
are removed, the ratio becomes 3 to 2. How many red
tomatoes were originally in the bag?
(A) 12 6. A particular integer N is divisible by two different prime
(B) 15 numbers p and q. Which of the following must be true?
(C) 18
(D) 24 I. N is not a prime number.
(E) 30
II. N is divisible by pq.
III. N is an odd integer.
2. If each digit in an integer is greater than the digit to the (A) I only
left, the integer is said to be “monotonic”. For example, (B) II only
12 is a monotonic integer since 2 > 1. How many positive (C) I and II only
two-digit monotonic integers are there? (D) I and III only
(E) I, II, and III
(A) 28
(B) 32
(C) 36
(D) 40 7. A perfect square is an integer that is the square of an
(E) 44 integer. Suppose that m and n are positive integers such
that mn > 15. If 15mn is a perfect square, what is the
least possible value of mn ?
a, 2a − 1, 3a − 2, 4a − 3, . . .
3. For a particular number a, the first term in the sequence
above is equal to a, and each term thereafter is 7 greater
than the previous term. What is the value of the 16th
term in the sequence? 8. M is a set of six consecutive even integers. When the
least three integers of set M are summed, the result is x.
When the greatest three integers of set M are summed,
the result is y. Which of the following is true?
(A) y = x − 18
3
4. If p is a prime number, how many factors does p have? (B) y = x + 18
(C) y = 2x
(D) y = 2x + 4
(A) One (E) y = 2x + 6
(B) Two
(C) Three
(D) Four
(E) Five
erikthered.com/tutor pg. 1
, SAT Math Hard Practice Quiz
9. A three-digit number, XYZ , is formed of three different 10. An integer is subtracted from its square. The result could
non-zero digits X, Y , and Z. A new number is formed by be which of the following?
rearranging the same three digits. What is the greatest
possible difference between the two numbers? (For ex-
ample, 345 could be rearranged into 435, for a difference (A) A negative integer.
of 435 − 345 = 90.) (B) An odd integer.
(C) The product of two consecutive even integers.
(D) The product of two consecutive odd integers.
(E) The product of two consecutive integers.
erikthered.com/tutor pg. 2
, SAT Math Hard Practice Quiz
Algebra and Functions 4. Let m and n be positive integers such that one-third of
m is n less than one-half of m. Which of the following is
a possible value of m ?
1. Let m be √
an even integer. How many possible values of
m satisfy m + 7 ≤ 3 ?
(A) 15
(B) 21
(A) One (C) 24
(B) Two (D) 26
(C) Three (E) 28
(D) Four
(E) Five
5. If a and b are numbers such that (a − 4)(b + 6) = 0, then
x+3 what is the smallest possible value of a2 + b2 ?
2. Let x be defined by x = for any x such that
x−1
x 6= 1. Which of the following is equivalent to x − 1 ?
x+2
(A)
x−1
4 6. Let f (x) = ax2 and g(x) = bx4 for any value of x. If a
(B) and b are positive constants, for how many values of x is
x−1
f (x) = g(x) ?
2x + 4
(C)
x−1
2 (A) None
(D) (B) One
x−1
(C) Two
x+2 (D) Three
(E)
x−2 (E) Four
3 2
3. Let a and b be numbers such √ that a = b . Which of the 7. Let a and b be numbers such that 30 < a < 40 and 50 <
following is equivalent to b a ?
b < 70. Which of the following represents all possible
values of a − b ?
(A) b2/3
(A) −40 < a − b < −20
(B) b4/3 (B) −40 < a − b < −10
(C) −30 < a − b < −20
(C) b2 (D) −20 < a − b < −10
(E) −20 < a − b < 30
(D) b3
(E) b4
erikthered.com/tutor pg. 3
