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Study Guides > Prealgebra

Identifying Multiples of Numbers

Learning Outcomes

  • Determine whether a given number is divisible by 2,3,5, or 10
Annie is counting the shoes in her closet. The shoes are matched in pairs, so she doesn’t have to count each one. She counts by twos: [latex]2,4,6,8,10,12[/latex]. She has [latex]12[/latex] shoes in her closet. The numbers [latex]2,4,6,8,10,12[/latex] are called multiples of [latex]2[/latex]. Multiples of [latex]2[/latex] can be written as the product of a counting number and [latex]2[/latex]. The first six multiples of [latex]2[/latex] are given below. [latex-display]\begin{array}{l}1\cdot 2=2\\ 2\cdot 2=4\\ 3\cdot 2=6\\ 4\cdot 2=8\\ 5\cdot 2=10\\ 6\cdot 2=12\end{array}[/latex-display] A multiple of a number is the product of the number and a counting number. So a multiple of [latex]3[/latex] would be the product of a counting number and [latex]3[/latex]. Below are the first six multiples of [latex]3[/latex]. [latex-display]\begin{array}{l}1\cdot 3=3\\ 2\cdot 3=6\\ 3\cdot 3=9\\ 4\cdot 3=12\\ 5\cdot 3=15\\ 6\cdot 3=18\end{array}[/latex-display] We can find the multiples of any number by continuing this process. The table below shows the multiples of [latex]2[/latex] through [latex]9[/latex] for the first twelve counting numbers.
Counting Number [latex]1[/latex] [latex]2[/latex] [latex]3[/latex] [latex]4[/latex] [latex]5[/latex] [latex]6[/latex] [latex]7[/latex] [latex]8[/latex] [latex]9[/latex] [latex]10[/latex] [latex]11[/latex] [latex]12[/latex]
[latex]\text{Multiples of }2[/latex] [latex]2[/latex] [latex]4[/latex] [latex]6[/latex] [latex]8[/latex] [latex]10[/latex] [latex]12[/latex] [latex]14[/latex] [latex]16[/latex] [latex]18[/latex] [latex]20[/latex] [latex]22[/latex] [latex]24[/latex]
[latex]\text{Multiples of }3[/latex] [latex]3[/latex] [latex]6[/latex] [latex]9[/latex] [latex]12[/latex] [latex]15[/latex] [latex]18[/latex] [latex]21[/latex] [latex]24[/latex] [latex]27[/latex] [latex]30[/latex] [latex]33[/latex] [latex]36[/latex]
[latex]\text{Multiples of }4[/latex] [latex]4[/latex] [latex]8[/latex] [latex]12[/latex] [latex]16[/latex] [latex]20[/latex] [latex]24[/latex] [latex]28[/latex] [latex]32[/latex] [latex]36[/latex] [latex]40[/latex] [latex]44[/latex] [latex]48[/latex]
[latex]\text{Multiples of }5[/latex] [latex]5[/latex] [latex]10[/latex] [latex]15[/latex] [latex]20[/latex] [latex]25[/latex] [latex]30[/latex] [latex]35[/latex] [latex]40[/latex] [latex]45[/latex] [latex]50[/latex] [latex]55[/latex] [latex]60[/latex]
[latex]\text{Multiples of }6[/latex] [latex]6[/latex] [latex]12[/latex] [latex]18[/latex] [latex]24[/latex] [latex]30[/latex] [latex]36[/latex] [latex]42[/latex] [latex]48[/latex] [latex]54[/latex] [latex]60[/latex] [latex]66[/latex] [latex]72[/latex]
[latex]\text{Multiples of }7[/latex] [latex]7[/latex] [latex]14[/latex] [latex]21[/latex] [latex]28[/latex] [latex]35[/latex] [latex]42[/latex] [latex]49[/latex] [latex]56[/latex] [latex]63[/latex] [latex]70[/latex] [latex]77[/latex] [latex]84[/latex]
[latex]\text{Multiples of }8[/latex] [latex]8[/latex] [latex]16[/latex] [latex]24[/latex] [latex]32[/latex] [latex]40[/latex] [latex]48[/latex] [latex]56[/latex] [latex]64[/latex] [latex]72[/latex] [latex]80[/latex] [latex]88[/latex] [latex]96[/latex]
[latex]\text{Multiples of }9[/latex] [latex]9[/latex] [latex]18[/latex] [latex]27[/latex] [latex]36[/latex] [latex]45[/latex] [latex]54[/latex] [latex]63[/latex] [latex]72[/latex] [latex]81[/latex] [latex]90[/latex] [latex]99[/latex] [latex]108[/latex]

Multiple of a Number

A number is a multiple of [latex]n[/latex] if it is the product of a counting number and [latex]n[/latex].
Recognizing the patterns for multiples of [latex]2, 5, 10, \text{and }3[/latex] will be helpful to you as you continue in this course. Doing the Manipulative Mathematics activity "Multiples" will help you develop a better understanding of multiples. The table below shows the counting numbers from [latex]1[/latex] to [latex]50[/latex]. Multiples of [latex]2[/latex] are highlighted. Do you notice a pattern? Multiples of [latex]2[/latex] between [latex]1[/latex] and [latex]50[/latex] The image shows a chart with five rows and ten columns. The first row lists the numbers from 1 to 10. The second row lists the numbers from 11 to 20. The third row lists the numbers from 21 to 30. The fourth row lists the numbers from 31 and 40. The fifth row lists the numbers from 41 to 50. All factors of 2 are highlighted in blue. The last digit of each highlighted number in the table is either [latex]0, 2, 4, 6, \text{or }8[/latex]. This is true for the product of [latex]2[/latex] and any counting number. So, to tell if any number is a multiple of [latex]2[/latex] look at the last digit. If it is [latex]0, 2, 4, 6, \text{or }8[/latex], then the number is a multiple of [latex]2[/latex].

example

Determine whether each of the following is a multiple of [latex]2\text{:}[/latex]
  1. [latex]489[/latex]
  2. [latex]3,714[/latex]
Solution:
1.
Is [latex]489[/latex] a multiple of [latex]2[/latex]?
Is the last digit [latex]0, 2, 4, 6, or 8[/latex]? No.
[latex]489[/latex] is not a multiple of [latex]2[/latex].
2.
Is [latex]3,714[/latex]a multiple of [latex]2[/latex]?
Is the last digit [latex]0, 2, 4, 6, or 8[/latex]? Yes.
[latex]3,714[/latex] is a multiple of [latex]2[/latex].
 

try it

#145413
The table below highlights the multiples of [latex]10[/latex] between [latex]1[/latex] and [latex]50[/latex]. All multiples of [latex]10[/latex] all end with a zero. Multiples of [latex]10[/latex] between [latex]1[/latex] and [latex]50[/latex] The image shows a chart with five rows and ten columns. The first row lists the numbers from 1 to 10. The second row lists the numbers from 11 to 20. The third row lists the numbers from 21 to 30. The fourth row lists the numbers from 31 and 40. The fifth row lists the numbers from 41 to 50. All factors of 10 are highlighted in blue.

example

Determine whether each of the following is a multiple of [latex]10\text{:}[/latex]
  1. [latex]425[/latex]
  2. [latex]350[/latex]

Answer: Solution:

1.
Is [latex]425[/latex] a multiple of [latex]10[/latex]?
Is the last digit [latex]0[/latex]? No.
[latex]425[/latex] is not a multiple of [latex]10[/latex].
2.
Is [latex]350[/latex] a multiple of [latex]10[/latex]?
Is the last digit [latex]0[/latex]? Yes.
[latex]350[/latex] is a multiple of [latex]10[/latex].

 

try it

#145418
Look back at the charts where you highlighted the multiples of [latex]2[/latex], of [latex]5[/latex], and of [latex]10[/latex]. Notice that the multiples of [latex]10[/latex] are the numbers that are multiples of both [latex]2[/latex] and [latex]5[/latex]. That is because [latex]10=2\cdot 5[/latex]. Likewise, since [latex]6=2\cdot 3[/latex], the multiples of [latex]6[/latex] are the numbers that are multiples of both [latex]2[/latex] and [latex]3[/latex]. The following video shows how to determine the first four multiples of 6. https://youtu.be/mkEWqspRVKk

Use Common Divisibility Tests

Another way to say that [latex]375[/latex] is a multiple of [latex]5[/latex] is to say that [latex]375[/latex] is divisible by [latex]5[/latex]. In fact, [latex]375\div 5[/latex] is [latex]75[/latex], so [latex]375[/latex] is [latex]5\cdot 75[/latex]. Notice in the last example that [latex]10,519[/latex] is not a multiple [latex]3[/latex]. When we divided [latex]10,519[/latex] by [latex]3[/latex] we did not get a counting number, so [latex]10,519[/latex] is not divisible by [latex]3[/latex].

Divisibility

If a number [latex]m[/latex] is a multiple of [latex]n[/latex], then we say that [latex]m[/latex] is divisible by [latex]n[/latex].
Since multiplication and division are inverse operations, the patterns of multiples that we found can be used as divisibility tests. The table below summarizes divisibility tests for some of the counting numbers between one and ten.
Divisibility Tests
A number is divisible by
[latex]2[/latex] if the last digit is [latex]0,2,4,6,\text{or }8[/latex]
[latex]3[/latex] if the sum of the digits is divisible by [latex]3[/latex]
[latex]5[/latex] if the last digit is [latex]5[/latex] or [latex]0[/latex]
[latex]6[/latex] if divisible by both [latex]2[/latex] and [latex]3[/latex]
[latex]10[/latex] if the last digit is [latex]0[/latex]

example

Determine whether [latex]1,290[/latex] is divisible by [latex]2,3,5,\text{and }10[/latex].

Answer: Solution: The table below applies the divisibility tests to [latex]1,290[/latex]. In the far right column, we check the results of the divisibility tests by seeing if the quotient is a whole number.

Divisible by…? Test Divisible? Check
[latex]2[/latex] Is last digit [latex]0,2,4,6,\text{or }8?[/latex] Yes. yes [latex]1290\div 2=645[/latex]
[latex]3[/latex] [latex]\text{Is sum of digits divisible by }3?[/latex] [latex]1+2+9+0=12[/latex] Yes. yes [latex]1290\div 3=430[/latex]
[latex]5[/latex] Is last digit [latex]5[/latex] or [latex]0?[/latex] Yes. yes [latex]1290\div 5=258[/latex]
[latex]10[/latex] Is last digit [latex]0?[/latex] Yes. yes [latex]1290\div 10=129[/latex]
Thus, [latex]1,290[/latex] is divisible by [latex]2,3,5,\text{and }10[/latex].

 

try it

#145433
 

example

Determine whether [latex]5,625[/latex] is divisible by [latex]2,3,5,\text{and }10[/latex].

Answer: Solution: The table below applies the divisibility tests to [latex]5,625[/latex] and tests the results by finding the quotients.

Divisible by…? Test Divisible? Check
[latex]2[/latex] Is last digit [latex]0,2,4,6,\text{or }8?[/latex] No. no [latex]5625\div 2=2812.5[/latex]
[latex]3[/latex] [latex]\text{Is sum of digits divisible by }3?[/latex] [latex]5+6+2+5=18[/latex] Yes. yes [latex]5625\div 3=1875[/latex]
[latex]5[/latex] Is last digit is [latex]5[/latex] or [latex]0?[/latex] Yes. yes [latex]5625\div 5=1125[/latex]
[latex]10[/latex] Is last digit [latex]0?[/latex] No. no [latex]5625\div 10=562.5[/latex]
Thus, [latex]5,625[/latex] is divisible by [latex]3[/latex] and [latex]5[/latex], but not [latex]2[/latex], or [latex]10[/latex].

 

try it

[ohm_question]145433[/ohm_question]
The following video lesson shows how to determine whether a number is divisible by [latex]2,3,4,5,6,8,9,10[/latex] https://youtu.be/i16N01IdIhk

Licenses & Attributions

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  • Question ID 145433, 145363, 145413, 145417, 145418. Authored by: Lumen Learnig. License: CC BY: Attribution. License terms: IMathAS Community License.

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  • Determine Multiples of a Given Number. Authored by: James Sousa (Mathispower4u.com). License: CC BY: Attribution.
  • Divisibility Rules. Authored by: James Sousa (Mathispower4u.com). License: CC BY: Attribution.

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