Factorise the following into their prime factors:
(a) 9271 (b) 2146 (c) 2 974 791
(a) $\lceil \sqrt{9271} \rceil = 97$, and $ 97^2 - 9271 = 138 $, which is not a perfect square.
$ 98^2 - 9271 = 333 $, which is not a perfect square.
$ 99^2 - 9271 = 530 $, which is not a perfect square.
$ 100^2 - 9271 = 27^2 $, and so $9271 = 100^2 - 27^2 = (100-27)(100+27) = 73 \times 127$.
The prime factorisation is $9271 = 73 \times 127$.
(b) Since 2146 is even, we can factor out 2, and work on the odd 1073.
$\lceil \sqrt{1073} \rceil = 33$, and $ 33^2 - 1073 = 4^2 $, and so $1073 = 33^2 - 4^2 = (33+4)(33-4) = 37 \times 29$.
The prime factorisation is $2146 = 2 \times 29 \times 37$.
(c) The Fermat factorisation approach doesn't yield easy results for several trials and so we reduce the number by factoring out 3, and working on 991597 as a smaller odd number.
$\lceil \sqrt{991597} \rceil = 996$, and $996^2 - 991597 = 419$ which is not a perfect square.
We keep trying until we find $1001^2 - 991597 = 102^2$, and so $991597 = 1001^2 - 102^2 = (1001 - 102)(1001 + 102) = 1103 \times 899 = 1103 \times 29 \times 31 $.
The prime factorisation is $2974791 = 3 \times 29 \times 31 \times 1103$.