Highest Common Factors

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Highest Common Factors

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A factor is a number that can be evenly divided into other whole numbers without having a remainder left over. For example, 9 is a common factor of 1, 3 and 9. Fractions and decimals are not found in common factors making finding common factors rather easy.

A common factor is a number that can be evenly divided into a collection or series of whole numbers. For example, the number 1 is a common number that can be evenly divided into any series of whole numbers.

Finding the factors of a whole number begins with starting with the number “1” and then dividing it with the given number. For example, let’s say that our given whole number is “9”.

9 ÷ 1 = 9

We see that when we divide “9” by “1” we get a whole number, i.e., “9”. We then continue to divide the number “9” with each ascending number as follows:

9 ÷ 2 = 4.5
9 ÷ 3 = 3
9 ÷ 4 = 2.25
9 ÷ 5 = 1.8
9 ÷ 6 = 1.5
9 ÷ 7 = 1.28
9 ÷ 8 = 1.12
9 ÷ 9 = 1

Once you divide the whole number into itself, this will give you the last factor. The last factor of 9 is “9”.

The factors of 9 are then, 1, 3 and 9 because the answer derived is a whole number. 2, 4, 5, 6, 7 and 8 are not factors because they did not divide evenly and the answer included either a fraction or decimal.

The highest common factor is when you take two or more whole numbers and then locate the highest common factor found in each number. For example, the highest factor of 9 is “9”. Now let’s compare that with the whole number 6.

6 ÷ 1 = 6
6 ÷ 2 = 3
6 ÷ 3 = 2
6 ÷ 4 = 1.5
6 ÷ 5 = 1.2
6 ÷ 6 = 1

The factors of 6 are 1, 2, 3 and 6. The highest factor of 6 is “6”. Now, which common factor of 6 matches the factors of 9? As we can see, both the numbers 1 and 3 are factors of the whole numbers 6 and 9. Now of the 1 and 3, which is the highest factor? The answer is “3”. 3 is the highest common factor.

1.
For the whole numbers given below, find the common highest factor.

15 and 20
1
4
5
6
15 ÷ 1 = 15; 15 ÷ 2 = 7.5; 15 ÷ 3 = 5; 15 ÷ 4 = 3.75; 15 ÷ 5 = 3; 15 ÷ 6 = 2.5; 15 ÷ 7 = 2.14; 15 ÷ 8 = 1.87; 15 ÷ 9 = 1.66; 15 ÷ 10 = 1.5; 15 ÷ 11 = 1.36; 15 ÷ 12 = 1.25; 15 ÷ 13 = 1.15; 15 ÷ 14 = 1.07; 15 ÷ 15 = 1

20 ÷ 1 = 20; 20 ÷ 2 = 10; 20 ÷ 3 = 6.66; 20 ÷ 4 = 5; 20 ÷ 5 = 4; 20 ÷ 6 = 3.33; 20 ÷ 7 = 2.85; 20 ÷ 8 = 2.5; 20 ÷ 9 = 2.22; 20 ÷ 10 = 2; 20 ÷ 11 = 1.81; 20 ÷ 12 = 1.66; 20 ÷ 13 = 1.53; 20 ÷ 14 = 1.42; 20 ÷ 15 = 1.33; 20 ÷ 16 = 1.25; 20 ÷ 17 = 1.17; 20 ÷ 18 = 1.11; 20 ÷ 19 = 1.05; 20 ÷ 20 = 1

The factors of 15 are 1, 3, 5 and 15. The factors of 20 are 1, 2, 4, 5, 10 and 20. The factors that can be found in both 15 and 20 include the numbers 1 and 5. 5 is, therefore, the highest common factor so Answer (b) is the correct answer
2.
For the whole numbers given below, find the common highest factor.

19 and 42
1
4
6
8
19 ÷ 1 = 19; 19 ÷ 2 = 9.5; 19 ÷ 3 = 6.33; 19 ÷ 4 = 4.75; 19 ÷ 5 = 3.8; 19 ÷ 6 = 3.16; 19 ÷ 7 = 2.71; 19 ÷ 8 = 2.37; 19 ÷ 9 = 2.11; 19 ÷ 10 = 1.9; 19 ÷ 11 = 1.72; 19 ÷ 12 = 1.58; 19 ÷ 13 = 1.46; 19 ÷ 14 = 1.35; 19 ÷ 15 = 1.26; 19 ÷ 16 = 1.18; 19 ÷ 17 = 1.11; 19 ÷ 18 = 1.05; 19 ÷ 19 = 1

42 ÷ 1 = 42; 42 ÷ 2 = 21; 42 ÷ 3 = 14; 42 ÷ 4 = 10.5; 42 ÷ 5 = 8.4; 42 ÷ 6 = 7; 42 ÷ 7 = 6; 42 ÷ 8 = 5.25; 42 ÷ 9 = 4.66; 42 ÷ 10 = 4.2; 42 ÷ 11 = 3.81; 42 ÷ 12 = 3.5; 42 ÷ 13 = 3.23; 42 ÷ 14 = 3; 42 ÷ 15 = 2.8; 42 ÷ 16 = 2.62; 42 ÷ 17 = 2.47; 42 ÷ 18 = 2.33; 42 ÷ 19 = 2.21; 42 ÷ 20 = 2.1; 42 ÷ 21 = 2; 42 ÷ 22 = 1.90; 42 ÷ 23 = 1.82; 42 ÷ 24 = 1.75; 42 ÷ 25 = 1.68; 42 ÷ 26 = 1.61; 42 ÷ 27 = 1.55; 42 ÷ 28 = 1.5; 42 ÷ 29 = 1.44; 42 ÷ 30 = 1.4; 42 ÷ 31 = 1.35; 42 ÷ 32 = 1.31; 42 ÷ 33 = 1.27; 42 ÷ 34 = 1.23; 42 ÷ 35 = 1.2; 42 ÷ 36 = 1.16; 42 ÷ 37 = 1.13; 42 ÷ 38 = 1.1; 42 ÷ 39 = 1.07; 42 ÷ 40 = 1.05; 42 ÷ 41 = 1.02; 42 ÷ 42 = 1

The factors of 19 are 1 and 19. The factors of 42 are 1, 2, 3, 6, 7, 14, 21 and 42. The factors that can be found in both 19 and 42 include only the number 1. 1 is, therefore, the highest common factor so Answer (a) is the correct answer
3.
For the whole numbers given below, find the common highest factor.

16 and 36
3
4
6
8
16 ÷ 1 = 16; 16 ÷ 2 = 8; 16 ÷ 3 = 5.33; 16 ÷ 4 = 4; 16 ÷ 5 = 3.2; 16 ÷ 6 = 2.66; 16 ÷ 7 = 2.57; 16 ÷ 8 = 2; 16 ÷ 9 = 1.77; 16 ÷ 10 = 1.6; 16 ÷ 11 = 1.45; 16 ÷ 12 = 1.33; 16 ÷ 13 = 1.23; 16 ÷ 14 = 1.14; 16 ÷ 15 = 1.06; 16 ÷ 16 = 1

36 ÷ 1 = 36; 36 ÷ 2 = 18; 36 ÷ 3 = 12; 36 ÷ 4 = 9; 36 ÷ 5 = 7.2; 36 ÷ 6 = 6; 36 ÷ 7 = 5.14; 36 ÷ 8 = 4.5; 36 ÷ 9 = 4; 36 ÷ 10 = 3.6; 36 ÷ 11 = 3.27; 36 ÷ 12 = 3; 36 ÷ 13 = 2.76; 36 ÷ 14 = 2.57; 36 ÷ 15 = 2.4; 36 ÷ 16 = 2.25; 36 ÷ 17 = 2.11; 36 ÷ 18 = 2; 36 ÷ 19 = 1.89; 36 ÷ 20 = 1.8; 36 ÷ 21 = 1.71; 36 ÷ 22 = 1.63; 36 ÷ 23 = 1.56; 36 ÷ 24 = 1.5; 36 ÷ 25 = 1.44; 36 ÷ 26 = 1.38; 36 ÷ 27 = 1.33; 36 ÷ 28 = 1.28; 36 ÷ 29 = 1.24; 36 ÷ 30 = 1.2; 36 ÷ 31 = 1.16; 36 ÷ 32 = 1.12; 36 ÷ 33 = 1.09; 36 ÷ 34 = 1.05; 36 ÷ 35 = 1.02; 36 ÷ 36 = 1

The factors of 16 are 1, 2, 4, 8 and 16. The factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18 and 36. The factors that can be found in both 16 and 36 include the numbers 1, 2 and 4. 4 is, therefore, the highest common factor so Answer (b) is the correct answer
4.
For the whole numbers given below, find the common highest factor.

24 and 32
8
12
16
24
24 ÷ 1 = 24; 24 ÷ 2 = 12; 24 ÷ 3 = 8; 24 ÷ 4 = 6; 24 ÷ 5 = 4.8; 24 ÷ 6 = 4; 24 ÷ 7 = 3.42; 24 ÷ 8 = 3; 24 ÷ 9 = 2.66; 24 ÷ 10 = 2.4; 24 ÷ 11 = 2.18; 24 ÷ 12 = 2; 24 ÷ 13 = 1.84; 24 ÷ 14 = 1.71; 24 ÷ 15 = 1.6; 24 ÷ 16 = 1.5; 24 ÷ 17 = 1.41; 24 ÷ 18 = 1.33; 24 ÷ 19 = 1.26; 24 ÷ 20 = 1.2; 24 ÷ 21 = 1.14; 24 ÷ 22 = 1.09; 24 ÷ 23 = 1.04; 24 ÷ 24 = 1

32 ÷ 1 = 32; 32 ÷ 2 = 16; 32 ÷ 3 = 10.66; 32 ÷ 4 = 8 ; 32 ÷ 5 = 6.4; 32 ÷ 6 = 5.33; 32 ÷ 7 = 4.57; 32 ÷ 8 = 4; 32 ÷ 9 = 3.55; 32 ÷ 10 = 3.2; 32 ÷ 11 = 2.90; 32 ÷ 12 = 2.66; 32 ÷ 13 = 2.46; 32 ÷ 14 = 2.28; 32 ÷ 15 = 2.13; 32 ÷ 16 = 2; 32 ÷ 17 = 1.88; 32 ÷ 18 = 1.77; 32 ÷ 19 = 1.68; 32 ÷ 20 = 1.6; 32 ÷ 21 = 1.52; 32 ÷ 22 = 1.45; 32 ÷ 23 = 1.39; 32 ÷ 24 = 1.33; 32 ÷ 25 = 1.28; 32 ÷ 26 = 1.23; 32 ÷ 27 = 1.18; 32 ÷ 28 = 1.14; 32 ÷ 29 = 1.1; 32 ÷ 30 = 1.06; 32 ÷ 31 = 1.03; 32 ÷ 32 = 1

The factors of 24 are 1, 2, 3, 4, 6, 8, 12 and 24. The factors of 32 are 1, 2, 4, 8, 16 and 32. The factors that can be found in both 24 and 32 include the numbers 1, 2, 4 and 8. 8 is, therefore, the highest common factor so Answer (a) is the correct answer
5.
For the whole numbers given below, find the common highest factor.

6 and 10
6
5
2
1
6 ÷ 1 = 6; 6 ÷ 2 = 3; 6 ÷ 3 = 2; 6 ÷ 4 = 1.5; 6 ÷ 5 = 1.2; 6 ÷ 6 = 1

10 ÷ 1 = 10; 10 ÷ 2 = 5; 10 ÷ 3 = 3.33; 10 ÷ 4 = 2.5; 10 ÷ 5 = 2; 10 ÷ 6 = 1.66; 10 ÷ 7 = 1.42; 10 ÷ 8 = 1.25; 10 ÷ 9 = 1.1; 10 ÷ 10 = 1

The factors of 6 are 1, 2, 3 and 6. The factors of 10 are 1, 2, 5 and 10. The factors that can be found in both 6 and 10 include the numbers 1 and 2. 2 is, therefore, the highest common factor so Answer (c) is the correct answer
6.
For the whole numbers given below, find the common highest factor.

13 and 26
1
3
6
13
13 ÷ 1 = 13; 13 ÷ 2 = 6.5; 13 ÷ 3 = 4.33; 13 ÷ 4 = 3.25; 13 ÷ 5 = 2.6; 13 ÷ 6 = 2.16; 13 ÷ 7 = 1.85; 13 ÷ 8 = 1.62; 13 ÷ 9 = 1.44; 13 ÷ 10 = 1.3; 13 ÷ 11 = 1.18; 13 ÷ 12 = 1.08; 13 ÷ 13 = 1

26 ÷ 1 = 26; 26 ÷ 2 = 13; 26 ÷ 3 = 8.66; 26 ÷ 4 = 6.5; 26 ÷ 5 = 5.2; 26 ÷ 6 = 4.33; 26 ÷ 7 = 3.71; 26 ÷ 8 = 3.25; 26 ÷ 9 = 2.88; 26 ÷ 10 = 2.6; 26 ÷ 11 = 2.36; 26 ÷ 12 = 2.16; 26 ÷ 13 = 2; 26 ÷ 14 = 1.85; 26 ÷ 15 = 1.73; 26 ÷ 16 = 1.62; 26 ÷ 17 = 1.52; 26 ÷ 18 = 1.44; 26 ÷ 19 = 1.36; 26 ÷ 20 = 1.3; 26 ÷ 21 = 1.23; 26 ÷ 22 = 1.18; 26 ÷ 23 = 1.13; 26 ÷ 24 = 1.08; 26 ÷ 25 = 1.04; 26 ÷ 26 = 1

The factors of 13 are 1 and 13. The factors of 26 are 1, 2, 13 and 26. The factors that can be found in both 13 and 26 include the numbers 1 and 13. 13 is, therefore, the highest common factor so Answer (d) is the correct answer
7.
For the whole numbers given below, find the common highest factor.

11 and 23
3
5
7
1
11 ÷ 1 = 11; 11 ÷ 2 = 5.5; 11 ÷ 3 = 3.66; 11 ÷ 4 = 2.75; 11 ÷ 5 = 2.2; 11 ÷ 6 = 1.83; 11 ÷ 7 = 1.57; 11 ÷ 8 = 1.37; 11 ÷ 9 = 1.22; 11 ÷ 10 = 1.1; 11 ÷ 11 = 1

23 ÷ 1 = 23; 23 ÷ 2 = 11.5; 23 ÷ 3 = 7.66; 23 ÷ 4 = 5.75; 23 ÷ 5 = 4.6; 23 ÷ 6 = 3.83; 23 ÷ 7 = 3.28; 23 ÷ 8 = 2.87; 23 ÷ 9 = 2.55; 23 ÷ 10 = 2.3; 23 ÷ 11 = 2.09; 23 ÷ 12 = 1.91; 23 ÷ 13 = 1.76; 23 ÷ 14 = 1.64; 23 ÷ 15 = 1.53; 23 ÷ 16 = 1.43; 23 ÷ 17 = 1.35; 23 ÷ 18 = 1.27; 23 ÷ 19 = 1.21; 23 ÷ 20 = 1.15; 23 ÷ 21 = 1.09; 23 ÷ 22 = 1.04; 23 ÷ 23 = 1

The factors of 11 are 1 and 11. The factors of 23 are 1 and 23. The factors that can be found in both 11 and 23 include only the number 1. 1 is, therefore, the highest common factor so Answer (d) is the correct answer
8.
For the whole numbers given below, find the common highest factor.

8 and 16
4
8
6
2
8 ÷ 1 = 8; 8 ÷ 2 = 4; 8 ÷ 3 = 2.66; 8 ÷ 4 = 2; 8 ÷ 5 = 1.6; 8 ÷ 6 = 1.33; 8 ÷ 7 = 1.14; 8 ÷ 8 = 1

16 ÷ 1 = 16; 16 ÷ 2 = 8; 16 ÷ 3 = 5.33; 16 ÷ 4 = 4; 16 ÷ 5 = 3.2; 16 ÷ 6 = 2.66; 16 ÷ 7 = 2.57; 16 ÷ 8 = 2; 16 ÷ 9 = 1.77; 16 ÷ 10 = 1.6; 16 ÷ 11 = 1.45; 16 ÷ 12 = 1.33; 16 ÷ 13 = 1.23; 16 ÷ 14 = 1.14; 16 ÷ 15 = 1.06; 16 ÷ 16 = 1

The factors of 8 are 1, 2, 4 and 8. The factors of 16 are 1, 2, 4, 8 and 16. The factors that can be found in both 8 and 16 include the numbers 1, 2, 4 and 8. 8 is, therefore, the highest common factor so Answer (b) is the correct answer
9.
For the whole numbers given below, find the common highest factor.

14 and 21
1
3
7
14
14 ÷ 1 = 14; 14 ÷ 2 = 7; 14 ÷ 3 = 4.66; 14 ÷ 4 = 3.5; 14 ÷ 5 = 2.8; 14 ÷ 6 = 2.33; 14 ÷ 7 = 2; 14 ÷ 8 = 1.74; 14 ÷ 9 = 1.55; 14 ÷ 10 = 1.4; 14 ÷ 11 = 1.27; 14 ÷ 12 = 1.16; 14 ÷ 13 = 1.07; 14 ÷ 14 = 1

21 ÷ 1 = 21; 21 ÷ 2 = 10.5; 21 ÷ 3 = 7; 21 ÷ 4 = 5.25; 21 ÷ 5 = 4.2; 21 ÷ 6 = 3.5; 21 ÷ 7 = 3; 21 ÷ 8 = 2.62; 21 ÷ 9 = 2.33; 21 ÷ 10 = 2.1; 21 ÷ 11 = 1.90; 21 ÷ 12 = 1.75; 21 ÷ 13 = 1.61; 21 ÷ 14 = 1.5; 21 ÷ 15 = 1.4; 21 ÷ 16 = 1.31; 21 ÷ 17 = 1.23; 21 ÷ 18 = 1.16; 21 ÷ 19 = 1.10; 21 ÷ 20 = 1.05; 21 ÷ 21 = 1

The factors of 14 are 1, 2, 7 and 14. The factors of 21 are 1, 3, 7 and 31. The factors that can be found in both 14 and 21 include the numbers 1 and 7. 7 is, therefore, the highest common factor so Answer (c) is the correct answer
10.
For the whole numbers given below, find the common highest factor.

9 and 18
9
3
1
4
9 ÷ 1 = 9; 9 ÷ 2 = 4.5; 9 ÷ 3 = 3; 9 ÷ 4 = 2.25; 9 ÷ 5 = 1.8; 9 ÷ 6 = 1.5; 9 ÷ 7 = 1.28; 9 ÷ 8 = 1.12; 9 ÷ 9 = 1

18 ÷ 1 = 18; 18 ÷ 2 = 9; 18 ÷ 3 = 6; 18 ÷ 4 = 4.5; 18 ÷ 5 = 3.6; 18 ÷ 6 = 3; 18 ÷ 7 = 2.57; 18 ÷ 8 = 2.25; 18 ÷ 9 = 2; 18 ÷ 10 = 1.8; 18 ÷ 11 = 1.63; 18 ÷ 12 = 1.5; 18 ÷ 13 = 1.38; 18 ÷ 14 = 1.28; 18 ÷ 15 = 1.2; 18 ÷ 16 = 1.12; 18 ÷ 17 = 1.05; 18 ÷ 18 = 1

The factors of 9 are 1, 3 and 9. The factors of 18 are 1, 2, 3, 6, 9 and 18. The factors that can be found in both 9 and 18 include the numbers 1, 3 and 9. 9 is, therefore, the highest common factor so Answer (a) is the correct answer
Author:  Christine G. Broome

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