Multiples Formula

The Formula

The k-th multiple of n is n \times k for k = 1, 2, 3, \ldots

When to use: Skip-counting produces multiples: counting by 3s gives 3, 6, 9, 12... — those are the multiples of 3.

Quick Example

Multiples of 4: 4, 8, 12, 16, 20 \ldots (4 \times 1, 4 \times 2, 4 \times 3 \ldots)

Notation

Multiples of n: \{n, 2n, 3n, 4n, \ldots\}

What This Formula Means

Numbers obtained by multiplying a given number by positive integers: the skip-counting sequence n, 2n, 3n, 4n, \ldots

Skip-counting produces multiples: counting by 3s gives 3, 6, 9, 12... — those are the multiples of 3.

Worked Examples

Example 1

easy
List the first 6 multiples of 7, and find the 20th multiple of 7.

Solution

  1. 1
    Multiples of 7: 7 \times 1 = 7, 7 \times 2 = 14, 7 \times 3 = 21, 7 \times 4 = 28, 7 \times 5 = 35, 7 \times 6 = 42.
  2. 2
    First 6 multiples: 7, 14, 21, 28, 35, 42.
  3. 3
    20th multiple: 7 \times 20 = 140.

Answer

First 6: 7, 14, 21, 28, 35, 42. The 20th multiple is 140.
Multiples of n are the values n, 2n, 3n, \ldots — the results of multiplying n by positive integers. The kth multiple is simply kn. There are infinitely many multiples of any non-zero integer.

Example 2

medium
Find all multiples of 6 between 50 and 100 (inclusive). How many are there?

Common Mistakes

  • Confusing multiples with factors — multiples of 5 are 5, 10, 15, 20... (going up), while factors of 20 are 1, 2, 4, 5, 10, 20 (dividing down)
  • Thinking multiples must be larger than the original number — the number itself is its smallest positive multiple (5 \times 1 = 5)
  • Listing non-multiples by adding instead of multiplying — the multiples of 7 are 7, 14, 21, 28\ldots (multiply by 1, 2, 3, 4\ldots), not 7, 8, 9, 10\ldots

Why This Formula Matters

Essential for finding common denominators (to add fractions), solving LCM problems, and understanding periodicity.

Frequently Asked Questions

What is the Multiples formula?

Numbers obtained by multiplying a given number by positive integers: the skip-counting sequence n, 2n, 3n, 4n, \ldots

How do you use the Multiples formula?

Skip-counting produces multiples: counting by 3s gives 3, 6, 9, 12... — those are the multiples of 3.

What do the symbols mean in the Multiples formula?

Multiples of n: \{n, 2n, 3n, 4n, \ldots\}

Why is the Multiples formula important in Math?

Essential for finding common denominators (to add fractions), solving LCM problems, and understanding periodicity.

What do students get wrong about Multiples?

Every number is its own smallest multiple (n = n \times 1); students sometimes think multiples must be strictly larger.

What should I learn before the Multiples formula?

Before studying the Multiples formula, you should understand: multiplication.

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