SSAT Middle Level

Factors, Multiples & Primes

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

Generate Unlimited Practice Questions

Learn This Topic

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 Middle Level

Q1 Medium

Which of the following numbers is divisible by 12?

A 314

B 412

C 516

D 622

E 736

Show Solution

Numbers that are divisible by 12 must be divisible by both 3 and 4. To be divisible by 4, the last two digits of the number must form a number divisible by 4. The numbers 412, 516, and 736 are all divisible by 4. To be divisible by 3, the sum of the digits must be divisible by 3. Let's check the sum of the digits for these three options: For 412, $4 + 1 + 2 = 7$ (not divisible by 3). For 516, $5 + 1 + 6 = 12$ (divisible by 3). For 736, $7 + 3 + 6 = 16$ (not divisible by 3). Therefore, only 516 is divisible by 12.

Answer: C

Q2 Medium

Mr. Davis has 48 pencils and 64 erasers. He wants to create identical supply packets for his students using all of the pencils and erasers, so that each packet has the same number of pencils and the same number of erasers. What is the greatest number of packets Mr. Davis can make?

A 4

B 8

C 12

D 16

E 24

Show Solution

To find the greatest number of identical packets that can be made without any leftover items, find the greatest common factor (GCF) of 48 and 64. The factors of 48 are 1, 2, 3, 4, 6, 8, 12, 16, 24, and 48. The factors of 64 are 1, 2, 4, 8, 16, 32, and 64. The greatest factor they share is 16. Therefore, Mr. Davis can make at most 16 packets (each containing 3 pencils and 4 erasers).

Answer: D

Q3 Medium

m

is an even number and

n

is an odd number, which of the following must be an odd number?

m + n + 1

m \times n

2 n + m

3 m + n

m (n + 2)

Show Solution

Let's evaluate the options using the rules of evens and odds. $m$ is even and $n$ is odd. In (D), $3 m$ is an odd times an even, which gives an even number. Then, adding the odd number $n$ to this even number results in an even plus an odd, which is always an odd number. Alternatively, you can plug in numbers: let $m = 2$ and $n = 1$ . Then $3 (2) + 1 = 6 + 1 = 7$ , which is odd.

Answer: D

Q4 Medium

What is the greatest prime factor of 130?

A 5

B 10

C 13

D 26

E 65

Show Solution

First, find the prime factorization of 130. Since 130 ends in 0, it is divisible by 10, so $130 = 10 \times 13$ . The prime factorization of 10 is $2 \times 5$ . So, the complete prime factorization of 130 is $2 \times 5 \times 13$ . The prime factors are 2, 5, and 13. The greatest of these prime factors is 13. (Note that while 26 and 65 are factors of 130, they are not prime numbers).

Answer: C

Q5 Medium

Two buses leave the station at the exact same time. Bus A completes its route and returns to the station every 20 minutes. Bus B completes its route and returns to the station every 35 minutes. How many minutes will it take for both buses to arrive at the station at the same time again?

A 55

B 70

C 100

D 140

E 700

Show Solution

To find when both buses will be at the station together again, find the least common multiple (LCM) of 20 and 35. The prime factorization of 20 is $2 \times 2 \times 5$ . The prime factorization of 35 is $5 \times 7$ . The LCM must include the highest power of each prime factor present: $(2 \times 2) \times 5 \times 7 = 4 \times 5 \times 7 = 140$ . They will meet again in 140 minutes.

Answer: D

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!

Generate Unlimited Practice Questions

Learn This Topic

Practice Questions

Tips & Strategies

Common Mistakes

Frequently Asked Questions

Related Topics

More Arithmetic & Number Sense

Other Levels

Browse

Generate Unlimited Practice Questions