🔢 Arithmetic — Topics

Master the core arithmetic topics that appear frequently in MATHCOUNTS problems.

Number Operations

Addition and Subtraction

Basic techniques:

  • Column addition: Line up digits by place value
  • Mental math: Use compatible numbers and shortcuts
  • Estimation: Round to check reasonableness

Common patterns:

  • Near multiples of 10: $47 + 19 = 47 + 20 - 1 = 66$
  • Complementary numbers: $47 + 53 = 100$
  • Breaking apart: $47 + 23 = (40 + 20) + (7 + 3) = 70$

Pitfall: Forgetting to carry or borrow Fix: Show work clearly and double-check each step

Multiplication and Division

Basic techniques:

  • Long multiplication: Multiply each digit systematically
  • Mental math: Use shortcuts and properties
  • Estimation: Round to check reasonableness

Common patterns:

  • Multiplying by powers of 10: Add zeros
  • Multiplying by 5: Multiply by 10, divide by 2
  • Multiplying by 25: Multiply by 100, divide by 4

Pitfall: Misplacing decimal points Fix: Count decimal places carefully

Number Properties

Even and Odd Numbers

Recognition: Check the last digit Properties:

  • Even + Even = Even
  • Odd + Odd = Even
  • Even + Odd = Odd
  • Even × Even = Even
  • Odd × Odd = Odd
  • Even × Odd = Even

Applications:

  • Parity problems: Determine if result is even or odd
  • Counting problems: Count even/odd numbers in a range
  • Algebra problems: Use properties to simplify expressions

Pitfall: Forgetting that 0 is even Fix: Remember that even means divisible by 2

Prime and Composite Numbers

Prime identification: Check divisibility by primes up to $\sqrt{n}$ First 25 primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97

Applications:

  • Factorization: Break numbers into prime factors
  • GCD/LCM: Find greatest common divisor and least common multiple
  • Number theory: Solve problems involving prime properties

Pitfall: Thinking 1 is prime Fix: Remember that primes have exactly two factors

Divisibility Rules

Memorize key rules:

  • 2: Last digit is even
  • 3: Sum of digits is divisible by 3
  • 4: Last two digits divisible by 4
  • 5: Last digit is 0 or 5
  • 6: Divisible by both 2 and 3
  • 9: Sum of digits is divisible by 9
  • 10: Last digit is 0

Applications:

  • Quick checks: Verify divisibility without division
  • Factorization: Find factors efficiently
  • Problem solving: Use rules to eliminate answer choices

Pitfall: Forgetting to check all digits Fix: Add digits systematically

Order of Operations

PEMDAS/BODMAS

Parentheses/Brackets: Do operations inside first Exponents/Orders: Calculate powers and roots Multiplication/Division: From left to right Addition/Subtraction: From left to right

Common mistakes:

  • Adding before multiplying: $2 + 3 \times 4 = 20$ (should be 14)
  • Subtracting before dividing: $8 - 4 \div 2 = 2$ (should be 6)
  • Forgetting parentheses: $2(3 + 4) = 14$ (not 10)

Fix: Use parentheses to clarify order

Mental Math Techniques

Shortcuts

Addition:

  • Adding 9: Add 10, subtract 1
  • Adding 11: Add 10, add 1
  • Adding near multiples of 10: Use compatible numbers

Subtraction:

  • Subtracting 9: Subtract 10, add 1
  • Subtracting 11: Subtract 10, subtract 1
  • Subtracting near multiples of 10: Use compatible numbers

Multiplication:

  • Multiplying by 5: Multiply by 10, divide by 2
  • Multiplying by 25: Multiply by 100, divide by 4
  • Multiplying by 11: Add digits and place in middle

Division:

  • Dividing by 5: Multiply by 2, divide by 10
  • Dividing by 25: Multiply by 4, divide by 100
  • Dividing by 4: Divide by 2 twice

Estimation

Front-end estimation: Use first digits Compatible numbers: Use easy-to-work-with numbers Benchmark estimation: Use known reference points

Applications:

  • Quick checks: Verify calculations
  • Problem solving: Narrow down answer choices
  • Time management: Skip exact calculations when appropriate

Number Sense

Place Value

Understanding: Each position represents a power of 10 Applications:

  • Rounding: Use place value to round correctly
  • Estimation: Use place value for quick estimates
  • Problem solving: Break numbers into parts

Rounding

Rules:

  • 0-4: Round down
  • 5-9: Round up
  • Ties: Round to even (banker’s rounding)

Applications:

  • Estimation: Round to make calculations easier
  • Problem solving: Use rounded numbers to check answers
  • Real-world problems: Round to appropriate precision

Scientific Notation

Format: $a \times 10^n$ where $1 \leq a < 10$ Examples:

  • $3,400 = 3.4 \times 10^3$
  • $0.0007 = 7 \times 10^{-4}$

Applications:

  • Large numbers: Express very large or very small numbers
  • Calculations: Simplify multiplication and division
  • Problem solving: Use scientific notation in word problems

Common Problem Types

Basic Calculations

Single operations: One arithmetic operation Multiple operations: Order of operations required Word problems: Translate words to arithmetic

Number Properties

Even/odd problems: Use properties to solve Prime problems: Use prime properties Divisibility problems: Use divisibility rules

Estimation Problems

Rounding: Round to specified place Approximation: Find approximate values Comparison: Compare numbers using estimation

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