Mathematical Reasoning

The SAE includes math problems that range from basic arithmetic to combinatorics and number theory. The key challenge is not just getting the right number — it's knowing when to reason step-by-step versus when to use code execution, and always returning the answer in the exact format requested.

Most SAE math answers expect a plain number with no explanation. If the question says "respond with only the number," your entire response should be something like 42 — nothing else.

When to Reason vs. When to Compute

Not all math is created equal. Some problems are best solved by reasoning; others demand precise computation.

Reason through it when:

The problem is conceptual ("How many ways can you arrange...")
The numbers are small enough to track mentally
The problem requires insight, not brute force

Use code execution when:

Large numbers are involved (factorials, exponents)
Precision matters (floating point, many decimal places)
The problem involves iteration or search
You need to verify a hand-calculated result

If the SAE environment provides code execution tools, use them for any computation beyond basic arithmetic. Mental math errors on large numbers are a common source of lost points.

Multi-Step Arithmetic

Many SAE questions chain several operations together. The key is careful bookkeeping.

Example: A store sells apples at $1.50 each and oranges at $2.00 each. You buy 7 apples and 5 oranges, then get a 10% discount. What do you pay?

Calculate subtotals

Apples: 7 x $1.50 = $10.50 Oranges: 5 x $2.00 = $10.00

Sum the subtotals

Total before discount: $10.50 + $10.00 = $20.50

Apply the discount

Discount: $20.50 x 0.10 = $2.05 Final: $20.50 - $2.05 = $18.45

If the question says "respond with only the number," the answer is 18.45.

Number Theory Basics

The SAE tests fundamental number theory concepts. Here are the ones that appear most often:

Prime Numbers

A number greater than 1 whose only divisors are 1 and itself. The first few: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29.

2 is the only even prime
1 is not prime (by convention)
To check if N is prime, test divisors up to the square root of N

GCD and LCM

GCD (Greatest Common Divisor): largest number dividing both. GCD(12, 8) = 4
LCM (Least Common Multiple): smallest number divisible by both. LCM(12, 8) = 24
Relationship: GCD(a,b) x LCM(a,b) = a x b

Divisibility Rules

Divisible by 2: last digit is even
Divisible by 3: digit sum is divisible by 3
Divisible by 5: last digit is 0 or 5
Divisible by 9: digit sum is divisible by 9

Checkpoint 1

Knowledge Check

What is the GCD of 48 and 36? Respond with only the number.

Combinatorics

Combinatorics questions ask "how many ways" something can happen. The two fundamental tools:

Permutations (order matters)

How many ways to arrange r items from n: P(n,r) = n! / (n-r)!

Example: How many 3-letter codes from 26 letters (no repeats)? P(26,3) = 26 x 25 x 24 = 15,600

Combinations (order doesn't matter)

How many ways to choose r items from n: C(n,r) = n! / (r! x (n-r)!)

Example: How many ways to choose 3 people from a group of 10? C(10,3) = 10! / (3! x 7!) = 120

When to Use Which

"Arrange," "order," "sequence," "first/second/third" = Permutation
"Choose," "select," "committee," "group" = Combination

Probability Basics

Probability questions on the SAE are usually straightforward:

P(event) = favorable outcomes / total outcomes

Example: A bag has 5 red, 3 blue, and 2 green marbles. What is the probability of drawing a blue marble?

P(blue) = 3/10 = 0.3

Watch out for:

"At least one" problems: P(at least one) = 1 - P(none)
Independent events: P(A and B) = P(A) x P(B)
Dependent events: P(A then B) = P(A) x P(B|A)

When the answer is a probability, check whether the question wants a fraction (3/10), a decimal (0.3), or a percentage (30%). The format matters.

Checkpoint 2