SSAT Upper Level

Factors, Multiples & Primes

Prime factorization, GCF, LCM, and divisibility rules — includes odd/even number properties

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Imagine you have a giant box of 24 cookies 🍪. If you want to share them equally with your friends, how many friends can you invite? You could invite 2, 3, 4, 6, 8, or 12 friends! These numbers are called factors—they are the numbers that divide perfectly into another number without leaving any crumbs (or remainders!). Factors are like the building blocks of numbers.

Now, imagine you are jumping on a giant trampoline. If you jump 3 feet every time, you will land on 3, 6, 9, 12, and so on. These landing spots are called multiples! Multiples are what you get when you multiply a number by 1, 2, 3, and keep going forever. 🦘

Finally, let's meet the VIPs of the number world: Prime numbers. A prime number is a number that only has two factors: 1 and itself. Think of them as super exclusive clubs where only the number 1 and the club owner are allowed inside! Numbers like 2, 3, 5, and 7 are primes. On the SSAT, you'll be a number detective 🕵️‍♂️, finding Greatest Common Factors (GCF) and Least Common Multiples (LCM) to solve puzzles. Don't worry, once you know the rules, it's easier than beating a video game level!

Key Formula

The Prime Factorization Tree is your best tool! Every number can be broken down into primes. For example,

24 = 2 \cdot 2 \cdot 2 \cdot 3

2^{3} \cdot 3

. Also, remember that finding the Greatest Common Factor (GCF) helps you simplify fractions! To simplify

\frac{12}{16}

, divide the top and bottom by their GCF (4) to get

\frac{3}{4}

Practice Questions

5 practice questions for SSAT Upper Level

Q1 Hard

x = 2^{4} \times 3^{2} \times 5 \times 13

and

y = 2^{2} \times 3^{3} \times 7^{2}

, what is the greatest common factor of

x

and

y

2 \times 3

2^{2} \times 3^{2}

2^{4} \times 3^{3}

2^{2} \times 3^{2} \times 5 \times 7^{2} \times 13

2^{4} \times 3^{3} \times 5 \times 7^{2} \times 13

Show Solution

To find the greatest common factor (GCF) of two numbers given in their prime factorization form, identify the prime factors that both numbers share and take the lowest power of each. The common prime factors of $x$ and $y$ are 2 and 3. The lowest power of 2 present in both is $2^{2}$ , and the lowest power of 3 present in both is $3^{2}$ . Therefore, the greatest common factor is $2^{2} \times 3^{2}$ .

Answer: B

Q2 Hard

p

represents an odd integer and

q

represents an even integer, which of the following must represent an odd integer?

p \times q

3 q + 2

2 p + q

3 p + 2 q

p^{2} + q^{2} + 1

Show Solution

We can determine the parity (odd/even) of each expression by applying the rules of arithmetic for odd and even numbers. We are given that $p$ is odd and $q$ is even.

(A) $p \times q$ : odd × even = even.
(B) $3 q + 2$ : 3(even) + even = even + even = even.
(C) $2 p + q$ : 2(odd) + even = even + even = even.
(D) $3 p + 2 q$ : 3(odd) + 2(even) = odd + even = odd.
(E) $p^{2} + q^{2} + 1$ : $(odd)^{2} + (even)^{2} + 1$ = odd + even + odd = even.
Therefore, $3 p + 2 q$ is the only expression that must result in an odd integer.

Answer: D

Q3 Hard

If 4X and 7Y represent 2-digit prime numbers, where X and Y are digits, what is the least possible value of

7 Y - 4 X

A 22

B 24

C 26

D 28

E 32

Show Solution

To find the least possible value of the difference $7 Y - 4 X$ , we need to make 7Y as small as possible and 4X as large as possible. The 2-digit prime numbers in the 40s are 41, 43, and 47. The 2-digit prime numbers in the 70s are 71, 73, and 79. The smallest possible value for 7Y is 71, and the largest possible value for 4X is 47. The least possible difference is $71 - 47 = 24$ .

Answer: B

Q4 Hard

x - y

represents a positive number that is a multiple of 5, which of the following must also be a multiple of 5?

x + y

x - 5 y

5 x - y

2 x - 2 y + 5

\frac{x - y}{5}

Show Solution

If $x - y$ is a multiple of 5, it can be written as $5 k$ for some integer $k$ . Let's evaluate the choices by looking for an expression that can be factored by 5. Choice (D) is $2 x - 2 y + 5$ . We can factor out a 2 from the first two terms to get $2 (x - y) + 5$ . Substituting $5 k$ for $(x - y)$ gives $2 (5 k) + 5 = 10 k + 5$ . This can be factored as $5 (2 k + 1)$ . Since the expression is 5 times an integer, it must be a multiple of 5.

Answer: D

Q5 Hard

A teacher has 90 pencils and 60 erasers to distribute to her students. She wants to create identical supply bags so that each bag contains the same number of pencils and the same number of erasers, with no supplies left over. If she creates the greatest possible number of supply bags, how many total items will be in each bag?

A 5

B 6

C 15

D 30

E 150

Show Solution

First, find the greatest number of supply bags the teacher can create by finding the greatest common factor (GCF) of 90 and 60. The prime factorization of 90 is $2 \times 3^{2} \times 5$ and for 60 is $2^{2} \times 3 \times 5$ . The GCF is $2 \times 3 \times 5 = 30$ , so she will create 30 bags. Next, determine the number of items in each bag. The number of pencils per bag is $90 \div 30 = 3$ . The number of erasers per bag is $60 \div 30 = 2$ . The total number of items in each bag is $3 + 2 = 5$ .

Answer: A

Tips & Strategies

Memorize your divisibility rules! Knowing that a number is divisible by 3 if its digits add up to a multiple of 3 will save you tons of time on the SSAT.
When a question asks about variables like 'a is a factor of n', plug in real numbers! Pick small, easy numbers like 2, 3, or 4 to test the answer choices.
Don't confuse factors and multiples! Factors are FEW (they are smaller than or equal to the number). Multiples are MANY (they are larger than or equal to the number).

Common Mistakes

Watch out for the number 1! A lot of students think 1 is a prime number, but it is NOT. A prime number must have exactly two distinct factors: 1 and itself.
Don't forget to check all prime factors when finding the GCF or LCM. It's easy to miss a $2$ or a $3$ if you don't draw your prime factorization tree neatly!

Frequently Asked Questions

What's the difference between a factor and a multiple?

Think of factors as the building blocks that make up a number (like 2 and 3 make 6). Multiples are what you get when you multiply that number by something else (like 6, 12, and 18 are multiples of 6).

Is 2 a prime number?

Yes! In fact, 2 is the ONLY even prime number. Since its only factors are 1 and 2, it gets to be in the exclusive prime number club.

How often do these questions appear on the SSAT?

You'll see several questions about factors, multiples, and primes on the SSAT Math sections. They love to test if you know your divisibility rules and how to find the LCM or GCF!

What is a Greatest Common Factor (GCF)?

The GCF is the biggest number that divides perfectly into two or more numbers. If you have 12 and 16, the biggest number that goes into both is 4!

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