ISEE Middle 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 Middle Level

Q1 Medium

Three buses depart from the same station at the same time. Bus A departs every 15 minutes, Bus B departs every 20 minutes, and Bus C departs every 25 minutes. How many minutes will it take for all three buses to depart from the station at the same time again?

A 60

B 150

C 300

D 7500

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To find when all three buses will leave together again, find the least common multiple (LCM) of 15, 20, and 25. Factor each number into its prime factorization: $15 = 3 \times 5$ , $20 = 2 \times 2 \times 5$ , $25 = 5 \times 5$ . The highest power of 2 present is $2 \times 2$ (from 20). The highest power of 3 present is 3 (from 15). The highest power of 5 present is $5 \times 5$ (from 25). Multiply these highest powers together: $2 \times 2 \times 3 \times 5 \times 5 = 300$ . It will take 300 minutes.

Answer: C

Q2 Medium

Which of the following represents the prime factorization of 126?

2 \times 7 \times 9

2 \times 3 \times 21

2 \times 3 \times 3 \times 7

6 \times 3 \times 7

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To find the prime factorization, break 126 down into its prime factors. First, divide by the smallest prime number, 2: $126 \div 2 = 63$ . Next, divide 63 by the prime number 3: $63 \div 3 = 21$ . Divide 21 by 3 again: $21 \div 3 = 7$ . Since 7 is a prime number, the factorization is complete. The prime factors are 2, 3, 3, and 7. Written as a product, this is $2 \times 3 \times 3 \times 7$ . Choices A, B, and D contain composite numbers (9, 21, and 6) and are not prime factorizations.

Answer: C

Q3 Medium

What is the greatest common factor of 36, 54, and 72?

A 6

B 9

C 12

D 18

Show Solution

Factor each number into its prime factorization: $36 = 2 \times 2 \times 3 \times 3$ , $54 = 2 \times 3 \times 3 \times 3$ , $72 = 2 \times 2 \times 2 \times 3 \times 3$ . To find the greatest common factor (GCF), identify the prime factors that all three numbers share. Each number has at least one 2 and two 3s. Multiply these shared factors together: $2 \times 3 \times 3 = 18$ . The greatest common factor is 18.

Answer: D

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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