Factors worksheet

1. Number of factors: 4

1. Break the number into primes. Every factor is built by picking how many of each prime to use, from none up to all of them

   $55$$=\text{5 × 11}$

2. Count the factors (d(n)): add 1 to each prime's power, since using none counts too, then multiply

   $\mathrm{d(n)}$$=\left(1+1\right)\times \left(1+1\right)$$=4$

3. Pair each small factor d (up to √55, the square root) with its partner 55 ÷ d. Every factor appears in exactly one pair

   $\text{1 × 55, 5 × 11}$

4. Add up all the factors (σ(n)) with a shortcut: one fraction per prime, multiplied together

   $\mathrm{\sigma (n)}$$=\frac{5^2-1}{5-1}\times \frac{{11}^2-1}{11-1}$$=72$

5. Leave out the number itself to get the sum of its proper factors

   $72-55$$=17$

2. Number of factors: 8

1. Break the number into primes. Every factor is built by picking how many of each prime to use, from none up to all of them

   $70$$=\text{2 × 5 × 7}$

2. Count the factors (d(n)): add 1 to each prime's power, since using none counts too, then multiply

   $\mathrm{d(n)}$$=\left(1+1\right)\times \left(1+1\right)\times \left(1+1\right)$$=8$

3. Pair each small factor d (up to √70, the square root) with its partner 70 ÷ d. Every factor appears in exactly one pair

   $\text{1 × 70, 2 × 35, 5 × 14, 7 × 10}$

4. Add up all the factors (σ(n)) with a shortcut: one fraction per prime, multiplied together

   $\mathrm{\sigma (n)}$$=\frac{2^2-1}{2-1}\times \frac{5^2-1}{5-1}\times \frac{7^2-1}{7-1}$$=144$

5. Leave out the number itself to get the sum of its proper factors

   $144-70$$=74$

3. Number of factors: 4

1. Break the number into primes. Every factor is built by picking how many of each prime to use, from none up to all of them

   $46$$=\text{2 × 23}$

2. Count the factors (d(n)): add 1 to each prime's power, since using none counts too, then multiply

   $\mathrm{d(n)}$$=\left(1+1\right)\times \left(1+1\right)$$=4$

3. Pair each small factor d (up to √46, the square root) with its partner 46 ÷ d. Every factor appears in exactly one pair

   $\text{1 × 46, 2 × 23}$

4. Add up all the factors (σ(n)) with a shortcut: one fraction per prime, multiplied together

   $\mathrm{\sigma (n)}$$=\frac{2^2-1}{2-1}\times \frac{{23}^2-1}{23-1}$$=72$

5. Leave out the number itself to get the sum of its proper factors

   $72-46$$=26$

4. Number of factors: 6

1. Break the number into primes. Every factor is built by picking how many of each prime to use, from none up to all of them

   $50$$=\text{2 × 5²}$

2. Count the factors (d(n)): add 1 to each prime's power, since using none counts too, then multiply

   $\mathrm{d(n)}$$=\left(1+1\right)\times \left(2+1\right)$$=6$

3. Pair each small factor d (up to √50, the square root) with its partner 50 ÷ d. Every factor appears in exactly one pair

   $\text{1 × 50, 2 × 25, 5 × 10}$

4. Add up all the factors (σ(n)) with a shortcut: one fraction per prime, multiplied together

   $\mathrm{\sigma (n)}$$=\frac{2^2-1}{2-1}\times \frac{5^3-1}{5-1}$$=93$

5. Leave out the number itself to get the sum of its proper factors

   $93-50$$=43$

5. Number of factors: 8

1. Break the number into primes. Every factor is built by picking how many of each prime to use, from none up to all of them

   $40$$=\text{2³ × 5}$

2. Count the factors (d(n)): add 1 to each prime's power, since using none counts too, then multiply

   $\mathrm{d(n)}$$=\left(3+1\right)\times \left(1+1\right)$$=8$

3. Pair each small factor d (up to √40, the square root) with its partner 40 ÷ d. Every factor appears in exactly one pair

   $\text{1 × 40, 2 × 20, 4 × 10, 5 × 8}$

4. Add up all the factors (σ(n)) with a shortcut: one fraction per prime, multiplied together

   $\mathrm{\sigma (n)}$$=\frac{2^4-1}{2-1}\times \frac{5^2-1}{5-1}$$=90$

5. Leave out the number itself to get the sum of its proper factors

   $90-40$$=50$

6. Number of factors: 7

1. Break the number into primes. Every factor is built by picking how many of each prime to use, from none up to all of them

   $64$$=\text{2⁶}$

2. Count the factors (d(n)): add 1 to each prime's power, since using none counts too, then multiply

   $\mathrm{d(n)}$$=\left(6+1\right)$$=7$

3. Pair each small factor d (up to √64, the square root) with its partner 64 ÷ d. One pair repeats a factor, because the number is a perfect square

   $\text{1 × 64, 2 × 32, 4 × 16, 8 × 8}$

4. Add up all the factors (σ(n)) with a shortcut: one fraction per prime, multiplied together

   $\mathrm{\sigma (n)}$$=\frac{2^7-1}{2-1}$$=127$

5. Leave out the number itself to get the sum of its proper factors

   $127-64$$=63$

7. Number of factors: 12

1. Break the number into primes. Every factor is built by picking how many of each prime to use, from none up to all of them

   $72$$=\text{2³ × 3²}$

2. Count the factors (d(n)): add 1 to each prime's power, since using none counts too, then multiply

   $\mathrm{d(n)}$$=\left(3+1\right)\times \left(2+1\right)$$=12$

3. Pair each small factor d (up to √72, the square root) with its partner 72 ÷ d. Every factor appears in exactly one pair

   $\text{1 × 72, 2 × 36, 3 × 24, 4 × 18, 6 × 12, 8 × 9}$

4. Add up all the factors (σ(n)) with a shortcut: one fraction per prime, multiplied together

   $\mathrm{\sigma (n)}$$=\frac{2^4-1}{2-1}\times \frac{3^3-1}{3-1}$$=195$

5. Leave out the number itself to get the sum of its proper factors

   $195-72$$=123$

8. Number of factors: 6

1. Break the number into primes. Every factor is built by picking how many of each prime to use, from none up to all of them

   $52$$=\text{2² × 13}$

2. Count the factors (d(n)): add 1 to each prime's power, since using none counts too, then multiply

   $\mathrm{d(n)}$$=\left(2+1\right)\times \left(1+1\right)$$=6$

3. Pair each small factor d (up to √52, the square root) with its partner 52 ÷ d. Every factor appears in exactly one pair

   $\text{1 × 52, 2 × 26, 4 × 13}$

4. Add up all the factors (σ(n)) with a shortcut: one fraction per prime, multiplied together

   $\mathrm{\sigma (n)}$$=\frac{2^3-1}{2-1}\times \frac{{13}^2-1}{13-1}$$=98$

5. Leave out the number itself to get the sum of its proper factors

   $98-52$$=46$

9. Number of factors: 6

1. Break the number into primes. Every factor is built by picking how many of each prime to use, from none up to all of them

   $45$$=\text{3² × 5}$

2. Count the factors (d(n)): add 1 to each prime's power, since using none counts too, then multiply

   $\mathrm{d(n)}$$=\left(2+1\right)\times \left(1+1\right)$$=6$

3. Pair each small factor d (up to √45, the square root) with its partner 45 ÷ d. Every factor appears in exactly one pair

   $\text{1 × 45, 3 × 15, 5 × 9}$

4. Add up all the factors (σ(n)) with a shortcut: one fraction per prime, multiplied together

   $\mathrm{\sigma (n)}$$=\frac{3^3-1}{3-1}\times \frac{5^2-1}{5-1}$$=78$

5. Leave out the number itself to get the sum of its proper factors

   $78-45$$=33$

10. Number of factors: 3

1. Break the number into primes. Every factor is built by picking how many of each prime to use, from none up to all of them

   $49$$=\text{7²}$

2. Count the factors (d(n)): add 1 to each prime's power, since using none counts too, then multiply

   $\mathrm{d(n)}$$=\left(2+1\right)$$=3$

3. Pair each small factor d (up to √49, the square root) with its partner 49 ÷ d. One pair repeats a factor, because the number is a perfect square

   $\text{1 × 49, 7 × 7}$

4. Add up all the factors (σ(n)) with a shortcut: one fraction per prime, multiplied together

   $\mathrm{\sigma (n)}$$=\frac{7^3-1}{7-1}$$=57$

5. Leave out the number itself to get the sum of its proper factors

   $57-49$$=8$