Example 12. Express the following as product of primes:

        (a) 2268 (b) 420

        Answers: (a) 2268 = 2² x 3⁴ x 7 (b) 420 = 2² x 3 x 5 x 7

        Divide by successive primes from 2 on: (a) 2268 = 2² x 3⁴ x 7
 

Factor / Divisor	2	2	3	3	3	3	7
Dividend	2268	1134	567	189	63	21	7
Quotient	1134	567	189	63	21	7	1
Remainder	0	0	0	0	0	0	0

    Divide by successive primes from 2 on: (a) 420 = 2² x 3 x 5 x 7
 

Factor / Divisor	2	2	3	5	7
Dividend	420	210	105	35	7
Quotient	210	105	35	7	1
Remainder	0	0	0	0	0

Example 13. Are the numbers (a) 598 and (b) 823 primes?

    (a) 

        Check to see if primes 2 to 23 is a divisor of 598

        or {2, 3, 5, 7, 11, 13, 17, 19, 23} | 598 ? Since answer is yes (13 | 598), 598 is not a prime

    (b) 

        Check to see if primes 2 to 23 is a divisor of 823

        or {2, 3, 5, 7, 11, 13, 17, 19, 23} | 598 ? Since answer is no, 823 is a prime.

Example 14. Use Number Theorems to state whether the following are divisible by:
    2, 3, 4, 5, 6, 10 or 11: 275, 78, 840, 896.
 

	2	3	4	5	6	8	9	10	11
275				Y					Y
78	Y	Y			Y
840	Y	Y	Y	Y	Y			Y
891		Y					Y		Y