1. Find all factors of the numbers below.

i) 256

256= 2x2x2x2x2x2x2x2

So we can write 256 as

2x128

4x64

8x32

16x16

1x256

Thus we  have nine factors

1,2,3,8,16,32,64,128,256

ii) 625

625= 5x5x5x5

So we can write 625 as

5x125

25x125

1x625

Then we have five factors

1,5,25,125,625

iii) 243

243= 3x3x3x3x3

So we can write 243 as

3x81

9x27

1x243

Thus we have six factors

1,3,9,27,81,243

iv) 121

121= 11x11

So we can write 121 as

11x11

1x121

Thus we have three factors 1,11,121

2. Find the number of factors of each of these numbers.

i) 500

ii) 600

iii) 700

iv) 800

v) 900

i) 500 = 2x2x5x5x5

(2x2) has 3 factors

(5x5x5) has 4 factors

500 has 3x4= 12 factors

ii) 600 = (2x2x2) x3x(5x5)

2x2x2 has 4 factors

3 has 2 factors

5x5 has 3 factors

therefore 600 has 4x2x3=24 factors

iii) 700 = (2x2) x(5x5)x7

2x2 has 3 factors

5x5 has 3 factors

7 has 2 factors

therefore 700 has 3x3x2= 18 factors

iv) 800 = (2x2x2x2x2)x5x5

2x2x2x2x2 has 6 factors

5x5 has 3 factors

800 has 6x3 = 18 factors

v) 900= 2x2x3x3x5x5

2x2 has 3 factors

3x3 has 3 factors

5x5 has 3 factors

900 has 3x3x3= 27 factors

3.How many factors does a product of three distinct primes have? What about a product of 4 distinct primes?

Each prime number has 2 factors. So the product of three distinct primes have 8 factors. The product of four distinct primes have 16 factors.

4. Find two numbers with exactly five factors.

Factors of 16 - 1, 2,4,8,16 (5 factors)

Product of 81 - 1,3,9,27,81 (5 factors)

Product of a prime number

(16 = 2x2x2x2;

81= 3x3x3x3x3)

ii) What is the smallest number with exactly five factors?

The smallest number with exactly five factors = 16

5. How many even factors does 3600 have ?

3600 has 36 factors.

(2,4,6,8,16,12,24,48,18,36,72,144,10,20,40,80,30,60,120,240,90,180,360,720,50,100,200,400,150,300,600,1200,450,900,1800,3600)