**Factors and Multiples**

**Factors**

**Factors** of any whole number are the set of those numbers which can divide the
whole number without leaving any remainder.

For example, 1, 2, 3, 4, 6, and 12 can divide 12 without leaving
any remainder, so 1, 2, 3, 4, 6, and 12 are the factors of 12.

We write this set of factors of 12 in symbolic form as given
below.

Factors of 12, F_{12} = {1, 2, 3, 4, 6, 12}

Similarly,

Factors of 1, F_{1} = {1}

Factors of 2, F_{2} = {1, 2}

Factors of 3, F_{3} = {1, 3}

Factors of 4, F_{4} = {1, 2, 4}

Factors of 5, F_{5} = {1, 5}

Factors of 6, F_{6} = {1, 2, 3, 6}

Factors of 7, F_{7} = {1, 7}

Factors of 8, F_{8} = {1, 2, 4, 8}

Factors of 9, F_{9} = {1, 3, 9}

Factors of 10, F_{10} = {1, 2, 5, 10}

Factors of 14, F_{12} = {1, 2, 7, 14}

Factors of 16, F_{16} = {1, 2, 4, 8, 16}

Factors of 18, F_{18} = {1, 2, 3, 6, 9,
18}

Factors of 20, F_{20} = {1, 2, 4, 5,
10, 20}

Factors of 24, F_{24} = {1, 2, 3, 4, 6,
8, 12, 24}

Factors of 28, F_{28} = { 1, 2, 4, 7,
14, 28}

Factors of 30, F_{30} = { 1, 2, 3, 5,
6, 10, 15, 30}

Factors of 32, F_{32} = { 1, 2, 4, 8,
16, 32}

Factors of 36, F_{36} = { 1, 2, 3, 4,
6, 9, 12, 18, 36}

Factors of 40, F_{40} = { 1, 2, 4, 5,
8, 10, 20, 40}

Factors of 48, F_{48} = { 1, 2, 3, 4,
6, 8, 12, 16, 24, 48}

Factors of 54, F_{54} = { 1, 2, 3, 6,
9, 18, 27, 54}

Factors of 60, F_{60} = { 1, 2, 3, 4,
5, 6, 10, 12, 15, 20, 30, 60}

Factors of 72, F_{72} = { 1, 2, 3, 4,
6, 8, 9, 12, 18, 24, 36, 72}

Factors of 75, F_{75} = { 1, 3, 5, 15,
25, 75}

Factors of 76, F_{76} = { 1, 2, 4, 19,
38, 76}

Factors of 78, F_{78} = { 1, 2, 3, 6,
13, 26, 39, 78}

Factors of 80, F_{80} = { 1, 2, 4, 5,
8, 10, 16, 20, 40, 80}

Factors of 84, F_{84} = { 1, 2, 3, 4,
6, 7, 12, 14, 21, 28, 42, 84}

Factors of 90, F_{90} = { 1, 2, 3, 5,
6, 9, 10, 15, 18, 30, 45, 90}

Factors of 100, F_{100} = { 1, 2, 4, 5,
10, 20, 25, 50, 100}

Factors of 128, F_{128} = { 1, 2, 4, 8,
16, 32, 64, 128}

**Multiples**

**Multiples** of any number are the set of numbers obtained by
multiplying that number with positive integers.

For example, multiples of 5 are
5, 10, 15, 20, 25, 30, 35, 40, 45, 50, ... etc.

We write this set of multiples of 5 in symbolic form as given
below.

M_{5} = {5, 10, 15, 20, 25, 30, 35, 40, 45, 50, …}

Similarly,

Multiples of 2, M_{2} = {2, 4, 6, 8, 10, 12, 14, 16, …}

Multiples of 3, M_{3} = {3, 6, 9, 12, 15, 18, 21, 24, …}

Multiples of 4, M_{4} = {4, 8, 12, 16, 20, 24, 28, 32, …}

Multiples of 6, M_{6} = {6, 12, 18, 24, 30, 36, 42, 48, …}

Multiples of 7, M_{7} = {7, 14, 21, 28, 35, 42, 49, 56, …}

Multiples of 8, M_{8} = {8, 16, 24, 32, 40, 48, 56, 64, …}

Multiples of 9, M_{9} = {9, 18, 27, 36, 45, 54, 63, 72, …}

Multiples of 10, M_{10} = {10, 20, 30, 40, 50, 60, 70,
80, …}

Multiples of 11, M_{11} = {11, 22, 33, 44, 55, 66, 77,
88, …}

Multiples of 12, M_{12} = {12, 24, 36, 48, 60, 72, 84,
96, …}

Multiples of 13, M_{13} = {13, 26, 39, 52, 65, 78, 91,
104, …}

Multiples of 14, M_{14} = {14, 28, 42, 56, 70, 84, 98,
112, …}

Multiples of 15, M_{15} = {15, 30, 45, 60, 75, 90, 105,
120, …}

Multiples of 16, M_{16} = {16, 32, 48, 64, 80, 96, 112,
128, …}

Multiples of 17, M_{17} = {17, 34, 51, 68, 85, 102, 119,
136, …}

Multiples of 18, M_{18} = {18, 36, 54, 72, 90, 108, 126,
144, …}

Multiples of 19, M_{19} = {19, 38, 57, 76, 95, 114, 133,
152, …}

Multiples of 20, M_{20} = {20, 40, 60, 80, 100, 120,
140, 160, …}

Multiples of 21, M_{21} = {21, 42, 63, 84, 105, 126,
147, 168, …}

Multiples of 22, M_{22} = {22, 44, 66, 88, 110, 132,
154, 176, …}

Multiples of 23, M_{23} = {23, 46, 69, 92, 115, 138,
161, 184, …}

Multiples of 24, M_{24} = {24, 48, 72, 96, 120, 144,
168, 192, …}

Multiples of 25, M_{25} = {25, 50, 75, 100, 125, 150,
175, 200, …}

**Worked Out Examples**

**Example 1:** Find the highest common factors 6 and 10.

**Solution: **

Here,

Factors of 6, F_{6} = {1, 2, 3, 6}

Factors
of 10, F_{10} = {1, 2, 5, 10}

∴ Common factors of 6 and 10 = {1, 2}

∴ The highest common factor of 6 and 10 = 2 Ans.

**Example 2:** Find the highest common factors of 12 and 16.

**Solution: **

Here,

Factors of 12, F_{12} = {1, 2, 3, 4, 6, 12}

Factors of 16, F_{16} = {1, 2, 4, 8, 16}

∴ Common factors
of 12 and 16 = {1, 2, 4}

∴ The highest common factor of 12 and 16 = 4 Ans.

**Example 3:** Find the highest common factors 20 and 30.

**Solution: **

Here,

Factors of 20, F_{20} = {1, 2, 4, 5,
10, 20}

Factors
of 30, F_{30} = { 1, 2, 3, 5, 6, 10, 15, 30}

∴ Common
factors of 20 and 30 = {1, 2, 5, 10}

∴ The highest common factor of 20 and 30 = 10 Ans.

**Example 4:** Find the highest common factors 40 and 50.

**Solution: **

Here,

Factors of 40, F_{40} = { 1, 2, 4, 5,
8, 10, 20, 40}

Factors
of 50, F_{50} = { 1, 2, 5, 10, 25, 50}

∴ Common
factors of 40 and 50 = {1, 2, 10}

∴ The highest common factor of 40 and 50 = 10 Ans.

**Example 5:** Find the highest common factors 54 and 72.

**Solution: **

Factors of 54, F_{54} = { 1, 2, 3, 6,
9, 18, 27, 54}

Factors
of 72, F_{72} = { 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72}

∴ Common
factors of 54 and 72 = {1, 2, 3, 6, 9, 18}

∴ The highest common factor of 54 and 72 = 18 Ans.

**Example 6:** Find the lowest common multiples 2 and 3.

**Solution: **

Here,

Multiples of 2, M_{2} = {2, 4, 6, 8, 10, 12, 14, 16, 18,
…}

Multiples of 3, M_{3}
= {3, 6, 9, 12, 15, 18, 21, 24, …}

∴ Common multiples of 2 and 3 = {6, 12, 18, …}

∴ A lowest common multiple of 2 and 3 = 6 Ans.

**Example 7:** Find the lowest common multiples of 4 and 5.

**Solution: **

Here,

Multiples of 4, M_{4} = {4, 8, 12, 16, 20, 24, 28, 32, 36,
40, …}

Multiples of 5, M_{5}
= {5, 10, 15, 20, 25, 30, 35, 40, …}

∴ Common multiples of 4 and 5 = {20, 40, …} Ans.

∴ A lowest common multiple of 4 and 5 = 20 Ans.

**Example 8:** Find the lowest common multiples 6 and 7.

**Solution: **

Here,

Multiples of 6, M_{6} = {6, 12, 18, 24, 30, 36, 42, 48,
54, 60, 66, 72, 78, 84 …}

Multiples of 7, M_{7}
= {7, 14, 21, 28, 35, 42, 49, 56, 63, 70, 77, 84, …}

∴ Common multiples of 6 and 7 = {42, 84, …}

∴ A lowest common multiple of 6 and 7 = 42 Ans.

**Example 9:** Find the lowest common multiples 10 and 15.

**Solution: **

Here,

Multiples of 10, M_{10} = {10, 20, 30, 40, 50, 60, 70,
80, 90, …}

Multiples of 15, M_{15}
= {15, 30, 45, 60, 75, 90, 105, 120, …}

∴ Common multiples of 10 and 15 = {30, 60, 90, …}

∴ A lowest common multiple of 10 and 15 = 30 Ans.

**Example 10:** Find the lowest common multiples 20 and 25.

**Solution: **

Here,

Multiples of 20, M_{20} = {20, 40, 60, 80, 100, 120,
140, 160, 180, 200, …}

Multiples of 25, M_{25}
= {25, 50, 75, 100, 125, 150, 175, 200, …}

∴ Common multiples of 20 and 25 = {100, 200, …}

∴ A lowest common multiple of 20 and 25 = 100 Ans.

If you have any questions or problems regarding the **Factors and Multiples**, you can ask here, in the comment section below.

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