Factors and Multiples

Free sample questions, a clear explanation, and 5 practice skills with an AI tutor that guides without giving the answer away.

Concept Review

Factor Pairs: The Math Mystery of Perfect Partners

Imagine you have 12 cookies and want to arrange them in neat rectangular rows on a baking sheet. How many different rectangles could you make? The answer lies in discovering the factor pairs of 12!

A factor pair is two numbers that multiply together to give you your target number. Think of them as perfect math partners — they work together to create something bigger.

Finding All the Partners

Let's solve the cookie mystery with 12. We need to find every pair of numbers that multiply to make 12:

Factor Pairs of 12:

1 × 12 = 12 → (1, 12)
2 × 6 = 12 → (2, 6)
3 × 4 = 12 → (3, 4)

Cookie Rectangles:

1 row of 12 cookies
2 rows of 6 cookies
3 rows of 4 cookies

Notice how we systematically check: Does 1 go into 12? Yes! Does 2 go into 12? Yes! Does 3 go into 12? Yes! We keep going until we've found them all.

🔍 The Factor Detective Trick

Here's the secret: factors always come in pairs! When you find one factor, you automatically discover its partner.

If 4 × 3 = 12, then both 4 and 3 are factors of 12. You get two factors for the price of one!

The Systematic Search

Let's try a trickier number: 24. Start with 1 and work your way up:

1 × 24 = 24 ✓
2 × 12 = 24 ✓
3 × 8 = 24 ✓
4 × 6 = 24 ✓
5 × ? = 24 ✗ (24 ÷ 5 = 4.8, not a whole number)

Factor pairs of 24: (1,24), (2,12), (3,8), (4,6)

🔑 Key Insight

Some numbers have lots of factor pairs (like 24 with 4 pairs), while others have very few. Prime numbers like 7 have exactly one factor pair: (1,7). The number of factor pairs tells you something special about how "divisible" a number is!

Key Takeaway: Those 12 cookies could be arranged in exactly 3 different rectangles because 12 has 3 factor pairs. Every whole number from 1 to 100 has its own unique collection of factor pairs — they're like mathematical fingerprints that reveal the hidden structure inside numbers.

Sample questions

1. Which number between 1 and 50 has the greatest number of factor pairs?

○ 47

✓ 48

○ 49

○ 50

Answer: 48 — 48 is highly composite! Its factors are 1, 2, 3, 4, 6, 8, 12, 16, 24, and 48. 47 is prime, and 49 only has three.

2. How can you use a rectangular area model to prove that 12 is a factor of 36?

○ By adding 12 and 36

○ By showing 12 is larger than 36

✓ By showing that 3 rectangles with side lengths of 12 can perfectly fit inside a 36-unit area

○ By drawing a circle

Answer: By showing that 3 rectangles with side lengths of 12 can perfectly fit inside a 36-unit area — Factors are the dimensions of rectangles that can be built with a specific total number of square units.

3. If a number has an odd number of factors (like 9, which has 1, 3, 9), what does that tell you about the number?

✓ It is a perfect square

○ It is a prime number

○ It is an even number

○ It is a multiple of 10

Answer: It is a perfect square — Square numbers have an odd number of factors because one factor (the square root) is multiplied by itself, so it isn't part of a "pair" of different numbers.

Skills in this topic

Find all factor pairs for a whole number in the range 1-100
Identify common factors of two whole numbers
Find the greatest common factor (GCF) of two numbers
List the first five multiples of a given 1-digit number
Identify common multiples and the least common multiple (LCM)

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