GMAT MATH ULTIMATE | 2026 UPDATED
STUDY GUIDE WITH PRACTICE QUESTIONS,
ANSWERS, AND RATIONALES
| GRADED A+ | GUARANTEED SUCCESS
Updated 2026 Questions and Answers
100% Verified Exam Prep and Comprehensive
Rationales Included
,3rd Rule of Probability: Conditional Probability Conditional Probability: The probability of event A AND event B occurring is the
probability of event A times the probability of event B, given that A has already
occurred.
P(A and B) = P(A) × P(B|A)
4th Rule of Probability: Probability of A OR B The probability of event A OR event B occurring is: the probability of event A
occurring plus the probability of event B occurring minus the probability of both
events occurring.
P(A or B) = P(A) + P(B) - P(A and B)
Probability of Multiple Events Rules:
• A and B < A or B
• A or B > Individual probabilities of A, B
• P(A and B) = P(A) x P(B) ← "fewer options"
• P(A or B) = P(A) + P(B) ← "more options"
Indistinguishable Events (i.e., anagrams with repeating To find the number of distinct permutations of a set of items with indistinguishable
letters) ("repeat") items, divide the factorial of the items in the set by the product of the
factorials of the number of indistinguishable elements.
Example: How many ways can the letters in TRUST be arranged? (5!)/(2!) = 60
5! is the factorial of items in the set, 2! is the factorial of the number of repeat
items ("T"s)
Combinations (Order Does Not Matter) nCr = n! / (r! (n - r)!)
Where n is the total number of items in the set and r is the number of chosen
items.
Permutations (Order Does Matter) nPr = n! / (n - r)!
Where n is the total number of items in the set and r is the number of chosen
items.
Circular Permutations The number of ways to arrange n distinct objects along a fixed circle is: (n - 1)!
Slope of a Line y = mx + b
m = slope = (difference in y coordinates)/(difference in x coordinates) = (y2 -
y1)/(x2-x1)
30-60-90 Triangle 30-60-90
x (shorter leg), x(sqrt 3) (longer leg), 2x (hypotenuse)
45-45-90 Triangle 45-45-90
x (shorter legs), x(sqrt 2) (hypotenuse)
Common Right Triangles 3-4-5 or 6-8-10 or 9-12-15
5-12-13
, Number Added or Deleted Use the mean to find number that was added or deleted.
• Total = mean x (number of terms)
• Number deleted = (original total) - (new total)
• Number added = (new total) - (original total)
Factors of Odd Numbers Odd numbers have only odd factors
Quadratic Formula To find roots of quadratic equation: ax^2+ bx + c = 0
x = [−b ± √(b^2 − 4ac)]/2a
Discriminant Quadratic equation: ax^2+ bx + c = 0
Dicriminant = b^2 - 4ac
If discriminiant > 0, there are two roots (and two x-intercepts)
If discriminant = 0, there is one root (and one x-intercept)
If discriminant < 0, there are no (real) roots
Exponents (x^r)(y^r)=(xy)^r
(3^3)(4^3)=12^3 = 1728
Prime Factorization: Greatest Common Factor (GCF) 1. Start by writing each number as product of its prime factors.
2. Write so that each new prime factor begins in same place.
3. Greatest Common Factor (GCF) is found by multiplying all factors appearing on
BOTH lists.
60 = 2 x 2 x 3 x 5
72 = 2 x 2 x 2 x 3 x 3
GCF = 2 x 2 x 3 = 12
Prime Factorization: Lowest Common Multiple (LCM) 1. Start by writing each number as product of its prime factors.
2. Write so that each new prime factor begins in same place.
3. Lowest common multiple found by multiplying all factors in EITHER list.
60 = 2 x 2 x 3 x 5
72 = 2 x 2 x 2 x 3 x 3
LCM = 2 x 2 x 2 x 3 x 3 x 5 = 360
Check for Prime 1. Pick a number n.
2. Start with the least prime number, 2. See if 2 is a factor of your number. If it is,
your number is not prime.
3. If 2 is not a factor, check to see if the next prime, 3, is a factor. If it is, your
number is not prime.
4. Keep trying the next prime number until you reach one that is a factor (in which
case n is not prime), or you reach a prime number that is equal to or greater than
the square root of n.
5. If you have not found a number less than or equal to the square root of n, you
can be sure that your number is prime.
Ex: the number n=19 has a square root of ~4.35. Test 2, 3, 4 --> you know 19 is prime
because none of them are factors, and any other factor would be greater than
sqrt(19).
STUDY GUIDE WITH PRACTICE QUESTIONS,
ANSWERS, AND RATIONALES
| GRADED A+ | GUARANTEED SUCCESS
Updated 2026 Questions and Answers
100% Verified Exam Prep and Comprehensive
Rationales Included
,3rd Rule of Probability: Conditional Probability Conditional Probability: The probability of event A AND event B occurring is the
probability of event A times the probability of event B, given that A has already
occurred.
P(A and B) = P(A) × P(B|A)
4th Rule of Probability: Probability of A OR B The probability of event A OR event B occurring is: the probability of event A
occurring plus the probability of event B occurring minus the probability of both
events occurring.
P(A or B) = P(A) + P(B) - P(A and B)
Probability of Multiple Events Rules:
• A and B < A or B
• A or B > Individual probabilities of A, B
• P(A and B) = P(A) x P(B) ← "fewer options"
• P(A or B) = P(A) + P(B) ← "more options"
Indistinguishable Events (i.e., anagrams with repeating To find the number of distinct permutations of a set of items with indistinguishable
letters) ("repeat") items, divide the factorial of the items in the set by the product of the
factorials of the number of indistinguishable elements.
Example: How many ways can the letters in TRUST be arranged? (5!)/(2!) = 60
5! is the factorial of items in the set, 2! is the factorial of the number of repeat
items ("T"s)
Combinations (Order Does Not Matter) nCr = n! / (r! (n - r)!)
Where n is the total number of items in the set and r is the number of chosen
items.
Permutations (Order Does Matter) nPr = n! / (n - r)!
Where n is the total number of items in the set and r is the number of chosen
items.
Circular Permutations The number of ways to arrange n distinct objects along a fixed circle is: (n - 1)!
Slope of a Line y = mx + b
m = slope = (difference in y coordinates)/(difference in x coordinates) = (y2 -
y1)/(x2-x1)
30-60-90 Triangle 30-60-90
x (shorter leg), x(sqrt 3) (longer leg), 2x (hypotenuse)
45-45-90 Triangle 45-45-90
x (shorter legs), x(sqrt 2) (hypotenuse)
Common Right Triangles 3-4-5 or 6-8-10 or 9-12-15
5-12-13
, Number Added or Deleted Use the mean to find number that was added or deleted.
• Total = mean x (number of terms)
• Number deleted = (original total) - (new total)
• Number added = (new total) - (original total)
Factors of Odd Numbers Odd numbers have only odd factors
Quadratic Formula To find roots of quadratic equation: ax^2+ bx + c = 0
x = [−b ± √(b^2 − 4ac)]/2a
Discriminant Quadratic equation: ax^2+ bx + c = 0
Dicriminant = b^2 - 4ac
If discriminiant > 0, there are two roots (and two x-intercepts)
If discriminant = 0, there is one root (and one x-intercept)
If discriminant < 0, there are no (real) roots
Exponents (x^r)(y^r)=(xy)^r
(3^3)(4^3)=12^3 = 1728
Prime Factorization: Greatest Common Factor (GCF) 1. Start by writing each number as product of its prime factors.
2. Write so that each new prime factor begins in same place.
3. Greatest Common Factor (GCF) is found by multiplying all factors appearing on
BOTH lists.
60 = 2 x 2 x 3 x 5
72 = 2 x 2 x 2 x 3 x 3
GCF = 2 x 2 x 3 = 12
Prime Factorization: Lowest Common Multiple (LCM) 1. Start by writing each number as product of its prime factors.
2. Write so that each new prime factor begins in same place.
3. Lowest common multiple found by multiplying all factors in EITHER list.
60 = 2 x 2 x 3 x 5
72 = 2 x 2 x 2 x 3 x 3
LCM = 2 x 2 x 2 x 3 x 3 x 5 = 360
Check for Prime 1. Pick a number n.
2. Start with the least prime number, 2. See if 2 is a factor of your number. If it is,
your number is not prime.
3. If 2 is not a factor, check to see if the next prime, 3, is a factor. If it is, your
number is not prime.
4. Keep trying the next prime number until you reach one that is a factor (in which
case n is not prime), or you reach a prime number that is equal to or greater than
the square root of n.
5. If you have not found a number less than or equal to the square root of n, you
can be sure that your number is prime.
Ex: the number n=19 has a square root of ~4.35. Test 2, 3, 4 --> you know 19 is prime
because none of them are factors, and any other factor would be greater than
sqrt(19).
