4th Grade Math Practice Test — Unit 6

Covers Unit 6 (Factors, multiples, and patterns). Use this reference to master factor pairs, multiples, prime and composite numbers, and pattern rules before taking the diagnostic test.

Open Study Guide: Free review reference with worked examples and check-yourself problems. Note: Achieving 90% mastery on Khan Academy beforehand is strongly recommended to prepare for test scoring.

Unit 6 — Factors, Multiples, and Patterns

Concept 1

Finding All Factor Pairs up to 100

The Big Idea: Factors are whole numbers that multiply together to make a target number. Factor pairs always come in twos: Number A × Number B = Target.

Systematic Rainbow / T-Chart Strategy: Always test numbers in order starting at 1:
  • Start with 1: 1 × [Target] always works.
  • Test 2: Is it an even number? If yes, divide by 2.
  • Test 3: Add the digits together (e.g. for 36: 3 + 6 = 9). If the sum is a multiple of 3, 3 is a factor.
  • Test 4: Can you halve the number twice?
  • Test 5: Does it end in 0 or 5?
  • Stop when factors cross over or repeat.

Worked Example: Find all factor pairs of 36

Test each number in order:

  • 1 × 36 = 36
  • 2 × 18 = 36
  • 3 × 12 = 36
  • 4 × 9 = 36
  • 5 does not divide evenly (ends in 6).
  • 6 × 6 = 36 (Stop here because 6 is paired with itself).

Factor Pairs: (1, 36), (2, 18), (3, 12), (4, 9), (6, 6).
List of Factors: 1, 2, 3, 4, 6, 9, 12, 18, 36 (9 unique factors).

Check Yourself:

Find all factor pairs of 48. How many unique factors does 48 have?

Show Answer & Explanation

Factor pairs of 48:
1 × 48 = 48
2 × 24 = 48
3 × 16 = 48
4 × 12 = 48
6 × 8 = 48
Total Unique Factors: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48 → 10 factors.

Understanding:
Concept 2

Multiples of Whole Numbers

The Big Idea: A multiple is what you get when you multiply a given number by 1, 2, 3, 4, 5, etc. Think of skip-counting!

Rule: Multiples are equal to or larger than the starting number (except 0). Every number is a multiple of itself!

Worked Example: Multiples of 8

Multiply 8 by counting numbers:

  • 8 × 1 = 8
  • 8 × 2 = 16
  • 8 × 3 = 24
  • 8 × 4 = 32
  • 8 × 5 = 40
  • 8 × 6 = 48

First 6 Multiples of 8: 8, 16, 24, 32, 40, 48.

Testing if 72 is a multiple of 8: Divide 72 by 8. Since 72 ÷ 8 = 9 with no remainder, yes, 72 is a multiple of 8.

Check Yourself:

Which of the following numbers are multiples of 7?
List: 14, 25, 35, 42, 50, 63

Show Answer & Explanation

Check each number with 7 times tables:
• 14 = 7 × 2 (Yes)
• 25 = not in 7s (No)
• 35 = 7 × 5 (Yes)
• 42 = 7 × 6 (Yes)
• 50 = not in 7s (No)
• 63 = 7 × 9 (Yes)
Answer: 14, 35, 42, 63.

Understanding:
Concept 3

Prime vs. Composite Numbers (1–100)

The Big Idea: Numbers can be grouped by how many factors they have:

  • Prime Number: Has exactly two factors: 1 and itself. (Examples: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31).
  • Composite Number: Has more than two factors. It can be broken into equal smaller groups. (Examples: 4, 6, 8, 9, 10, 12, 14, 15).
  • The Number 1: Has only one factor (1 × 1). It is NEITHER prime NOR composite!
Common 4th Grade Traps:
  • 2 is the ONLY even prime number! Every other even number is composite because 2 is a factor.
  • Not all odd numbers are prime! 9 = 3 × 3, 15 = 3 × 5, 21 = 3 × 7, 27 = 3 × 9, 51 = 3 × 17, 91 = 7 × 13. Always test 3 and 7 before deciding an odd number is prime.

Worked Example: Classify 29 and 51

Is 29 prime or composite?
Factors of 29: Only 1 × 29. No other factors divide evenly.
→ 29 is PRIME.

Is 51 prime or composite?
Digits: 5 + 1 = 6 (multiple of 3). That means 3 × 17 = 51.
Factors: 1, 3, 17, 51 (4 factors).
→ 51 is COMPOSITE.

Check Yourself:

Classify each number as Prime, Composite, or Neither:
1, 2, 39, 67, 91

Show Answer & Explanation

• 1: Neither prime nor composite (only 1 factor).
• 2: Prime (only 1 × 2).
• 39: Composite (3 × 13 = 39).
• 67: Prime (only 1 × 67).
• 91: Composite (7 × 13 = 91).

Understanding:
Concept 4

Distinguishing Factors vs. Multiples

The Big Idea: Students often swap the words factor and multiple. Keep this mental picture:

  • Factors are SMALLER (or equal): They fit inside the number.
    [Factor] × [Factor] = Target Number
  • Multiples are LARGER (or equal): They grow out of the number.
    Target Number × [Count] = [Multiple]

Worked Example: Relate 6 and 24

• 6 is a factor of 24, because 6 × 4 = 24 (6 fits into 24).
• 24 is a multiple of 6, because 6 × 4 = 24 (24 grows out of 6).

Check Yourself:

Find a number between 20 and 30 that is a multiple of both 3 and 4.

Show Answer & Explanation

Multiples of 3 between 20 and 30: 21, 24, 27.
Multiples of 4 between 20 and 30: 24, 28.
The common multiple is 24.

Understanding:
Concept 5

Number Sequence Patterns & Rules

The Big Idea: A pattern follows an exact rule from term to term. You need to identify the rule, generate the terms, and notice special features (such as all numbers being odd, or alternating even/odd).

Worked Example: Rule “Start at 4, add 6”

Start with 4, then keep adding 6:

  • Term 1: 4
  • Term 2: 4 + 6 = 10
  • Term 3: 10 + 6 = 16
  • Term 4: 16 + 6 = 22
  • Term 5: 22 + 6 = 28

Observation Feature: All terms end in 4, 0, 6, 2, 8 and all terms are even numbers (Even + Even is always Even).

Check Yourself:

Rule: “Start at 3, multiply by 2 then subtract 1.”
Find the first 4 terms of the sequence.

Show Answer & Explanation

• Term 1: 3
• Term 2: (3 × 2) − 1 = 6 − 1 = 5
• Term 3: (5 × 2) − 1 = 10 − 1 = 9
• Term 4: (9 × 2) − 1 = 18 − 1 = 17
Terms: 3, 5, 9, 17.

Understanding:
Concept 6

Word Problems with Arrays & Grouping

The Big Idea: When real-world items must be split into equal rows or groups with no leftovers, the group size must be a factor of the total count.

Worked Example: Organizing 40 students

A gym teacher wants to organize 40 students into equal teams with no students left out.

• Can she make teams of 6? 40 ÷ 6 = 6 R4. No, 4 students are left over.
• Can she make teams of 5? 40 ÷ 5 = 8 R0. Yes, exactly 8 teams of 5 students.
• Other possible team sizes (factors of 40): 1, 2, 4, 5, 8, 10, 20, 40.

Check Yourself:

An art teacher has 54 tiles to arrange into a solid rectangle with no tiles left over.
Can the rectangle be 4 tiles wide? If not, what is one possible width greater than 5?

Show Answer & Explanation

54 ÷ 4 = 13 R2 → No, cannot be 4 tiles wide.
Factors of 54: 1, 2, 3, 6, 9, 18, 27, 54.
Possible widths greater than 5: 6, 9, 18, 27, or 54 tiles wide.

Understanding: