📐 Concept diagram

00-01 — Whole Number Arithmetic

Phase: 0 — Arithmetic & Number Foundations Subject: 00-01 Prerequisites: None (starting point — Year 8 level) Next subject: 00-02 — Fractions

Learning Objectives

By the end of this subject, you will be able to:

1. Perform addition, subtraction, multiplication, and division with whole numbers confidently, including with large numbers
2. Apply the order of operations (PEMDAS/BODMAS) correctly to evaluate multi-step arithmetic expressions
3. Execute long division and interpret remainders
4. Identify factors, multiples, primes, and composites, and compute prime factorizations
5. Find the Greatest Common Factor (GCF) and Least Common Multiple (LCM) of two or more numbers

Core Content

1. Place Value

Every digit in a whole number has a place value determined by its position. Each position is a power of 10:

$... billions  hundred-millions  ten-millions  millions  hundred-thousands  ten-thousands  thousands  hundreds  tens  ones
$

For example, in 73,582:

The number is: 70,000 + 3,000 + 500 + 80 + 2 = 73,582.

Place value is why we "carry" in addition and "borrow" in subtraction — you're moving values between columns.

2. Addition

Addition combines two or more numbers into a sum.

Algorithm (column addition): 1. Write numbers aligned by place value (units under units, tens under tens, etc.) 2. Add each column from right to left 3. If a column sum is ≥ 10, write the units digit and carry the tens digit to the next column

Example: Calculate 4,827 + 3,596

$   ¹ ¹ ¹     ← carries
   4 8 2 7
+  3 5 9 6
-----------
   8 4 2 3
$

• Column 1 (ones): 7 + 6 = 13 → write 3, carry 1
• Column 2 (tens): 1 + 2 + 9 = 12 → write 2, carry 1
• Column 3 (hundreds): 1 + 8 + 5 = 14 → write 4, carry 1
• Column 4 (thousands): 1 + 4 + 3 = 8 → write 8

Result: 8,423

Properties of addition: - Commutative: a + b = b + a (order doesn't matter) - Associative: (a + b) + c = a + (b + c) (grouping doesn't matter) - Identity: a + 0 = a (adding zero changes nothing)

3. Subtraction

Subtraction finds the difference between two numbers. Formally: c = a − b means b + c = a.

Algorithm (column subtraction with borrowing): 1. Write numbers aligned by place value (larger number on top) 2. Subtract each column from right to left 3. If the top digit is smaller than the bottom digit, borrow from the next left column

Example: Calculate 6,205 − 3,847

$   ⁵ ¹¹ ⁹
   6 2 0 5    ← strike through and rewrite after borrowing
−  3 8 4 7
-----------
   2 3 5 8
$

Let's walk through the borrowing:

• Column 4 (ones): 5 − 7 → can't. Borrow from tens. But tens is 0, so borrow from hundreds. But hundreds is 0, so borrow from thousands.
• Borrow 1 from thousands (6→5). Add 10 to hundreds (0→10).
• Now borrow 1 from hundreds (10→9). Add 10 to tens (0→10).
• Now borrow 1 from tens (10→9). Add 10 to ones (5→15).
• 15 − 7 = 8
• Column 3 (tens): 9 − 4 = 5
• Column 2 (hundreds): 9 − 8 = 1
• Column 1 (thousands): 5 − 3 = 2

Result: 2,358

Check your answer: 2,358 + 3,847 = 6,205 ✓

Subtraction is not commutative: a − b ≠ b − a (unless a = b).

4. Multiplication

Multiplication is repeated addition: 5 × 3 = 5 + 5 + 5 = 15.

Algorithm (long multiplication): 1. Write the larger number on top 2. Multiply the top number by each digit of the bottom number, starting from the right 3. Each partial product shifts one place left 4. Add all partial products

Example: Calculate 427 × 358

$        4 2 7
    ×   3 5 8
    ---------
      3 4 1 6    ← 427 × 8
    2 1 3 5 0    ← 427 × 5, shifted left 1 place
  1 2 8 1 0 0    ← 427 × 3, shifted left 2 places
  -----------
  1 5 2 8 6 6
$

Step-by-step for the first partial product (427 × 8): - 8 × 7 = 56 → write 6, carry 5 - 8 × 2 + 5 = 16 + 5 = 21 → write 1, carry 2 - 8 × 4 + 2 = 32 + 2 = 34 → write 34 - Partial product: 3,416

Properties of multiplication: - Commutative: a × b = b × a - Associative: (a × b) × c = a × (b × c) - Distributive over addition: a × (b + c) = a × b + a × c - Identity: a × 1 = a - Zero property: a × 0 = 0

⚠️ THIS IS CRITICAL — The distributive property is the foundation of all algebra. You will use it thousands of times in later subjects (expanding brackets, factoring, linear algebra). Make sure you understand it deeply now.

5. Division

Division splits a number (the dividend) into equal parts determined by the divisor. The result is the quotient.

Formally: a ÷ b = c means b × c = a. Division is the inverse of multiplication.

If a is not evenly divisible by b, there is a remainder r where 0 ≤ r < b: a = b × q + r (q is quotient, r is remainder)

Algorithm (long division):

Example: Calculate 8,736 ÷ 24

$        3 6 4
      --------
24 ) 8 7 3 6
      7 2       ← 24 × 3 = 72 (how many 24s in 87?)
      ---
      1 5 3     ← bring down 3
      1 4 4     ← 24 × 6 = 144 (how many 24s in 153?)
      -----
          9 6   ← bring down 6
          9 6   ← 24 × 4 = 96 (how many 24s in 96?)
          ---
            0
$

Result: 364 (no remainder)

Algorithm steps: 1. Divide: How many times does the divisor go into the current working digits? 2. Multiply: Multiply the divisor by that digit 3. Subtract: Subtract from the working digits 4. Bring down: Bring down the next digit of the dividend 5. Repeat until all digits are brought down

6. Order of Operations (PEMDAS / BODMAS)

When an expression has multiple operations, you must evaluate them in a specific order:

Key rules: - Multiplication and division have equal precedence — work from left to right - Addition and subtraction have equal precedence — work from left to right - Operations inside parentheses are evaluated first (innermost parentheses first)

Example: Evaluate 10 + 6 × (8 − 3)² ÷ 5

10 + 6 × (8 − 3)² ÷ 5
= 10 + 6 × 5² ÷ 5         ← Parentheses: (8 − 3) = 5
= 10 + 6 × 25 ÷ 5          ← Exponents: 5² = 25
= 10 + 150 ÷ 5             ← Multiplication (left): 6 × 25 = 150
= 10 + 30                  ← Division: 150 ÷ 5 = 30
= 40                       ← Addition

Common pitfall: People who ignore order of operations get: 10 + 6 × (8 − 3)² ÷ 5 → 16 × (5)² ÷ 5 → 16 × 25 ÷ 5 → 400 ÷ 5 = 80 (WRONG!)

The mistake was adding 10 + 6 first instead of doing the multiplication first.

7. Factors and Multiples

A factor of a number n is a whole number that divides n exactly (without remainder).

b is a factor of n if n ÷ b has remainder 0.

A multiple of a number n is the product of n and any whole number: n × 1, n × 2, n × 3, ...

m is a multiple of n if m = n × k for some whole number k.

Examples: - Factors of 12: 1, 2, 3, 4, 6, 12 (because 12 ÷ 1, 12 ÷ 2, 12 ÷ 3, 12 ÷ 4, 12 ÷ 6, 12 ÷ 12 all have remainder 0) - Multiples of 7: 7, 14, 21, 28, 35, 42, 49, ...

Relationship: If a is a factor of b, then b is a multiple of a.

8. Prime and Composite Numbers

A prime number is a whole number > 1 that has exactly two factors: 1 and itself.

A composite number is a whole number > 1 that has more than two factors.

The number 1 is neither prime nor composite — it has exactly one factor (itself).

⚠️ THIS IS CRITICAL — Primes are the building blocks of all whole numbers. Prime factorization appears repeatedly in later subjects including cryptography (RSA), algebraic number theory, and modular arithmetic.

First few primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, ...

Note: 2 is the only even prime — every other even number is divisible by 2 and therefore composite.

9. Prime Factorization

Every composite number can be expressed as a unique product of prime numbers. This is the Fundamental Theorem of Arithmetic.

Method (factor tree):

Example: Find the prime factorization of 84

$         84
        /  \
       2   42
          /  \
         2   21
            /  \
           3    7
$

Working from the bottom up: 84 = 2 × 2 × 3 × 7 = 2² × 3 × 7

Systematic method: 1. Start with the smallest prime (2). Divide repeatedly while possible. 2. Move to the next prime (3, 5, 7, ...) and repeat. 3. Continue until the result is 1.

$84 ÷ 2 = 42
42 ÷ 2 = 21
21 ÷ 3 = 7
7 ÷ 7 = 1

84 = 2 × 2 × 3 × 7 = 2² × 3 × 7
$

10. Greatest Common Factor (GCF)

The GCF (also called GCD — Greatest Common Divisor) of two or more numbers is the largest number that divides all of them exactly.

Method using prime factorization: 1. Find the prime factorization of each number 2. For each common prime, take the smallest exponent 3. Multiply these together

Example: Find GCF(48, 180)

48 = 2⁴ × 3
180 = 2² × 3² × 5

Common primes: 2 and 3
Smallest exponent for 2: 2²
Smallest exponent for 3: 3¹

GCF = 2² × 3 = 4 × 3 = 12

Check: 48 ÷ 12 = 4, 180 ÷ 12 = 15 ✓

11. Least Common Multiple (LCM)

The LCM of two or more numbers is the smallest number that is a multiple of all of them.

Method using prime factorization: 1. Find the prime factorization of each number 2. For each prime that appears, take the largest exponent 3. Multiply these together

Example: Find LCM(48, 180)

48 = 2⁴ × 3
180 = 2² × 3² × 5

All primes: 2, 3, 5
Largest exponent for 2: 2⁴
Largest exponent for 3: 3²
Largest exponent for 5: 5¹

LCM = 2⁴ × 3² × 5 = 16 × 9 × 5 = 720

Check: 720 ÷ 48 = 15, 720 ÷ 180 = 4 ✓

Important relationship: For any two numbers a and b:

GCF(a,b) × LCM(a,b) = a × b

For 48 and 180: 12 × 720 = 8,640 = 48 × 180 ✓

This relationship gives you a shortcut: if you've found the GCF, you can get the LCM by dividing the product by the GCF.

Key Terms

• Addition and subtraction
• Division
• Fundamental Theorem of Arithmetic
• Multiplication
• Place value
• Prime numbers
• Repeat

Worked Examples

Example 1: Multi-Step Arithmetic with Order of Operations

Evaluate: (24 ÷ 8 + 3²) × (15 − 7) + 4

Solution:

$(24 ÷ 8 + 3²) × (15 − 7) + 4
= (24 ÷ 8 + 9) × 8 + 4        ← Exponents (3² = 9) and parentheses (15 − 7 = 8)
= (3 + 9) × 8 + 4             ← Division inside first parentheses: 24 ÷ 8 = 3
= 12 × 8 + 4                  ← Addition inside parentheses: 3 + 9 = 12
= 96 + 4                      ← Multiplication: 12 × 8 = 96
= 100                         ← Addition
$

Answer: 100

Example 2: Long Division with Remainder

Calculate: 15,283 ÷ 37. Give quotient and remainder.

Solution:

$         4 1 3  r 2
       -------------
37 ) 1 5 2 8 3
      1 4 8         ← 37 × 4 = 148, 152 − 148 = 4
      -----
          4 8       ← bring down next digit (8), so we have 48
          3 7       ← 37 × 1 = 37, 48 − 37 = 11
          ---
          1 1 3     ← bring down next digit (3), so we have 113
          1 1 1     ← 37 × 3 = 111, 113 − 111 = 2
          -----
              2     ← remainder
$

Answer: Quotient = 413, Remainder = 2 Check: 37 × 413 + 2 = 15,281 + 2 = 15,283 ✓

Example 3: Prime Factorization and GCF/LCM

Find the GCF and LCM of 126 and 294.

Solution:

Step 1 — Prime factorization:

$126 ÷ 2 = 63
63 ÷ 3 = 21
21 ÷ 3 = 7
7 ÷ 7 = 1
→ 126 = 2 × 3² × 7

294 ÷ 2 = 147
147 ÷ 3 = 49
49 ÷ 7 = 7
7 ÷ 7 = 1
→ 294 = 2 × 3 × 7²
$

Step 2 — GCF: Take each common prime with the smallest exponent.

$Common primes: 2¹, 3¹, 7¹
GCF = 2 × 3 × 7 = 42
$

Step 3 — LCM: Take all primes with the largest exponent.

$2¹, 3², 7²
LCM = 2 × 9 × 49 = 882
$

Verification: 42 × 882 = 37,044 = 126 × 294 ✓

Answer: GCF = 42, LCM = 882

Practice Problems

(Answers are below. Try each problem before checking.)

Problem 1: Calculate 53,891 + 27,406

Problem 2: Calculate 80,003 − 46,789

Problem 3: Calculate 6,248 × 507

Problem 4: Calculate 45,932 ÷ 26. Give quotient and remainder.

Problem 5: Evaluate: 36 ÷ (11 − 2) × 2 + 5² − 3 × 4

Problem 6: Find the prime factorization of 360.

Problem 7: Find the GCF and LCM of 84 and 120.

Summary

1. Place value is the foundation of all arithmetic — each digit represents a power of 10
2. Addition and subtraction use column alignment with carrying and borrowing
3. Multiplication uses partial products shifted by place value; the distributive property ⚠️ (a × (b + c) = a × b + a × c) is essential for all algebra
4. Division is the inverse of multiplication; long division handles large numbers systematically
5. Order of operations (PEMDAS/BODMAS) determines evaluation order: Parentheses → Exponents → Multiplication/Division (L→R) → Addition/Subtraction (L→R)
6. Prime numbers ≥ 2 with exactly two factors are the building blocks of all whole numbers — every composite has a unique prime factorization
7. GCF uses the smallest exponents of common primes; LCM uses the largest exponents of all primes; GCF × LCM = a × b

Pitfalls

• Ignoring order of operations. Adding before multiplying is the most common mistake — 10 + 6 × 5 is 40, not 80. Always follow PEMDAS/BODMAS.
• Misidentifying 1 as prime. 1 has exactly one factor (itself), not two. It is neither prime nor composite. Many students miscount the factors.
• Mixing up GCF and LCM. GCF takes the smallest exponent of common primes; LCM takes the largest exponent of all primes. Swapping these produces wrong answers.
• Forgetting to carry or borrow correctly. In column addition, the carried digit must be added to the next column; in subtraction, borrowing across zeros (e.g., 6005 − 3847) requires chaining across multiple columns.
• Losing track of place value in long multiplication. Each partial product must shift one place left. Misaligning the partial products (or forgetting the trailing zeros) is a persistent source of error.

Quiz

Answer each question, then read the explanation for your choice.

Q1: Evaluate: 8 + 2 × (7 − 3)²

A) 24 B) 40 C) 72 D) 160

Q2: What is 10,005 − 4,837?

A) 5,168 B) 6,168 C) 5,178 D) 6,178

Q3: What is the prime factorization of 180, expressed with exponents?

A) 2 × 2 × 3 × 3 × 5 B) 2² × 3 × 5² C) 2² × 3² × 5 D) 2 × 3² × 10

Q4: Find the GCF of 72 and 108.

A) 12 B) 18 C) 24 D) 36

Q5: What is the LCM of 24 and 36?

A) 36 B) 72 C) 144 D) 864

Q6: When performing the division 53,742 ÷ 39, what is the quotient and remainder?

A) 1,378 remainder 0 B) 1,377 remainder 39 C) 1,378 remainder 20 D) 1,377 remainder 21

Q7: Which of the following is a prime number?

A) 51 B) 57 C) 61 D) 87

Q8: Evaluate: (40 ÷ 5 + 2³) × (6 − 1) + 10 ÷ 2

A) 40 B) 45 C) 85 D) 50

Next Steps

Move on to 00-02 — Fractions to learn about proper/improper fractions, equivalent fractions, simplifying, and fraction operations.

Q5: Which of the following numbers is prime?

A) 51 B) 57 C) 61 D) 87

Answer: C) 61

51 = 3 × 17, 57 = 3 × 19, 87 = 3 × 29. 61 is not divisible by any prime ≤ √61 ≈ 7.8 (tested 2, 3, 5, 7 — none divide 61).