Factorise the following integers into their prime factors:
(a) 299 (b) 851 (c) 10403 (d) 2479
We will use Fermat difference of two squares factorisation method.
(a) $\lceil \sqrt(299) \rceil = 18$, and $18^2-299 = 5^2$.
So $299 = 18^2 - 5^2 = (18+5)(18-5) = 23 \times 13$.
So the prime factorisation is $299=23 \times 13$.
(b) $\lceil \sqrt(851) \rceil = 30$, and $30^2-851 = 7^2$.
So $851 = 30^2 - 7^2 = (30+7)(30-7) = 37 \times 23$.
So the prime factorisation is $851 = 37 \times 23$.
(c) $\lceil \sqrt(10403) \rceil = 102$, and $102^2-10403 = 1^2$.
So $10403 = (102^2+1)(102^2-1) = 103 \times 101$.
So the prime factorisation is $10403 = 103 \times 101$.
(d) $\lceil \sqrt(2479) \rceil = 50$, and $50^2-2479 = 21$ which is not a perfect square.
So we try $51^2-2479=122$, also not a perfect square.
So we try $52^2 - 2479 = 15^2$, and so $2479 = 52^2 - 15^2 = (52+15)(52-15)= 67 \times 37$.
So $2479 = 67 \times 37$.