ISEE Upper Level

Factors, Multiples & Primes

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

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Imagine you have 12 cupcakes. How can you arrange them evenly? You could do 1 row of 12, 2 rows of 6, or 3 rows of 4. Those numbers—1, 2, 3, 4, 6, and 12—are the Factors of 12! Factors are the numbers that multiply together to make a bigger number. Think of them as the building blocks of math! 🧁 Remember that a factor divides perfectly into the whole, so if 3 is a factor of 12, the fraction $\frac{3}{12}$ simplifies perfectly to $\frac{1}{4}$ .

Now, what if you're scoring 3-pointers in basketball? Your score goes 3, 6, 9, 12, 15... Those are Multiples. Multiples are what you get when you count by a number. They multiply and get bigger and bigger! 🏀

Finally, meet the VIPs of the number world: Prime Numbers. A prime number is a number greater than 1 that only has two factors: 1 and itself. Think of the number 7. You can't break it into even groups. It only hangs out with 1 and 7! The opposite of a prime number is a composite number, which has lots of factors.

On the ISEE, you'll see these concepts a lot in the Quantitative Reasoning and Math Achievement sections. Knowing your factors, multiples, and primes will help you solve puzzles fast! Remember, there's no penalty for guessing on the ISEE, so if you're ever stuck, take your best shot! 🎯

Key Formula

To find the Least Common Multiple (LCM) or Greatest Common Factor (GCF), use Prime Factorization! Break a number down until only prime numbers are left. Example:

12 = 2 \cdot 2 \cdot 3

2^{2} \cdot 3

Practice Questions

3 practice questions for ISEE Upper Level

Q1 Hard

x

and

y

are distinct prime numbers, what is the greatest common factor of

15 x^{2} y

and

10 x y^{3}

5 x y

5 x^{2} y^{3}

30 x y

30 x^{2} y^{3}

Show Solution

To find the greatest common factor (GCF) of algebraic terms, find the GCF of the numerical coefficients and then the GCF of the variables by taking the lowest power of each common prime variable. The GCF of the coefficients 15 and 10 is 5. The lowest power of $x$ between $x^{2}$ and $x$ is $x$ . The lowest power of $y$ between $y$ and $y^{3}$ is $y$ . Multiplying these together gives a greatest common factor of $5 x y$ .

Answer: A

Q2 Hard

p

and

q

are distinct prime numbers, what is the least common multiple of

9 p^{2} q

12 p q^{2}

, and

6 p^{3}

3 pq

36 p^{2} q^{2}

36 p^{3} q^{2}

72 p^{3} q^{2}

Show Solution

To find the least common multiple (LCM) of algebraic terms, find the LCM of the numerical coefficients and multiply it by the highest power of each prime variable present. The LCM of the coefficients 9, 12, and 6 is 36. The highest power of $p$ among $p^{2}$ , $p$ , and $p^{3}$ is $p^{3}$ . The highest power of $q$ among $q$ , $q^{2}$ , and $1$ (no $q$ term) is $q^{2}$ . Multiplying these together gives a least common multiple of $36 p^{3} q^{2}$ .

Answer: C

Q3 Hard

x

is a multiple of

y

, and

y

is a prime number, what is the least common multiple of

x

and

y

x

y

x y

\frac{x}{y}

Show Solution

The least common multiple (LCM) of two numbers is the smallest positive integer that is a multiple of both numbers. We are given that $x$ is already a multiple of $y$ . Every number is also a multiple of itself ( $x = x \times 1$ ), which means $x$ is a multiple of both $x$ and $y$ . Therefore, the least common multiple of $x$ and $y$ is simply $x$ . The fact that $y$ is a prime number is extra information that does not change this relationship.

Answer: A

Tips & Strategies

Use the plug-in trick! 🧩 If an ISEE question uses letters (variables) for factors and multiples, pick small, easy numbers to test the answer choices.
Memorize your divisibility rules! Knowing that numbers ending in 0 or 5 are divisible by 5, or that an even number is always divisible by 2, saves tons of time on the Math Achievement section. ⏱️
When finding the LCM, always start by listing the multiples of the largest number first. It gets you to the answer much faster!

Common Mistakes

Watch out for the number 1! 🚨 A lot of students think 1 is a prime number, but it's not. The smallest prime number is 2, which is also the only even prime number!
Don't mix up Factors and Multiples! Remember: Factors are few (smaller building blocks), and Multiples are many (they multiply and get bigger forever). 🧱

Frequently Asked Questions

Do I need to memorize all the prime numbers for the ISEE?

You don't need to memorize all of them, but knowing the prime numbers under 20 (2, 3, 5, 7, 11, 13, 17, 19) will give you a huge speed boost on test day! 🚀

What is a Quantitative Comparison question?

On the ISEE, you'll see questions asking you to compare Column A and Column B. For factors and multiples, you might have to compare the GCF of two numbers against the LCM of two numbers. Just calculate both sides and pick the bigger one! ⚖️

What if I can't find the factors of a really big number?

Don't panic! Use a factor tree to break it down step by step. Start by dividing by 2 if it's even, or 5 if it ends in 0 or 5. 🌲

Should I guess if I don't know the GCF or LCM?

Yes! There is absolutely zero penalty for wrong answers on the ISEE. If you're stuck, eliminate any choices that look obviously wrong and pick your favorite letter! ✨

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