Factors and Multiples Practice Quiz with a Step-by-Step Interactive Lesson

Factors and factor pairs

Key idea

A factor of a number \(n\) is a whole number that divides \(n\) evenly. In other words, \(a\) is a factor of \(n\) if \(n \div a\) has no remainder. Factors come in pairs: if \(a\) is a factor of \(n\), then there is a number \(b\) such that \(a\times b = n\).

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Summary

• A factor divides a number evenly (no remainder).
• Use factor pairs to list all factors in order.

Multiples and counting multiples

Key idea

A multiple of \(n\) is a number you get by multiplying \(n\) by a whole number: \(n\times 1, n\times 2, n\times 3,\dots\). If \(m\) is a multiple of \(n\), then \(n\) is a factor of \(m\).

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Summary

• Multiples are made by multiplying by whole numbers: \(n,2n,3n,\dots\).
• To count multiples up to a limit, you can list them or use division carefully.

Prime numbers, composite numbers, and prime factorization

Key idea

A prime number has exactly two positive factors: \(1\) and itself. A composite number has more than two positive factors. The number \(1\) is neither prime nor composite.

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Summary

• Prime: exactly two factors. Composite: more than two factors.
• Prime factorization writes a composite as a product of primes (for example, \(27=3^3\)).

Common factors and the greatest common factor (GCF)

Key idea

Common factors are factors shared by two numbers. The greatest common factor (GCF) is the largest factor they share. You can find it by listing factors or by using prime factorization.

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Summary

• Common factors are shared factors.
• The GCF is the largest common factor.

Common multiples and the least common multiple (LCM)

Key idea

Common multiples are multiples shared by two or more numbers. The least common multiple (LCM) is the smallest positive number that is a multiple of each number. You can find it by listing multiples or using prime factorization.

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Summary

• Common multiples are shared multiples.
• The LCM is the smallest positive common multiple.

Divisibility rules to test factors and multiples quickly

Key idea

Divisibility rules help you decide quickly if a number divides evenly. Here are some common rules:

• Divisible by 2: last digit is even (0,2,4,6,8).
• Divisible by 3: sum of digits is divisible by 3.
• Divisible by 4: last two digits form a number divisible by 4.
• Divisible by 5: last digit is 0 or 5.
• Divisible by 6: divisible by 2 and by 3.
• Divisible by 9: sum of digits is divisible by 9.
• Divisible by 10: last digit is 0.

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Summary

• Divisibility rules help you test quickly without long division.
• Use them to decide if a number is a factor or a multiple.

Why factors and multiples matter

Where you use factors and multiples

• Simplifying: use the GCF to simplify fractions and ratios.
• Common denominators: use the LCM to add and subtract fractions with different denominators.
• Schedules: repeating events meet again after an LCM amount of time.
• Arrays and rectangles: factor pairs describe possible row-and-column arrangements.

Worked example: repeating events (LCM)

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Final recap

• Factors divide evenly; multiples come from multiplying by whole numbers.
• Use factor pairs to list factors and keep them in order.
• Prime vs. composite: prime has exactly two factors; composite has more than two.
• GCF is the biggest shared factor; LCM is the smallest shared multiple.
• Divisibility rules help you check quickly and reduce mistakes.