Anonymous

try dividing by prime numbers, starting at 3. A long process. Or find a website that will do it for you, with answer 271 x 331

Anonymous

As a product of its prime factors: 271 times 331 = 89,701

Anonymous

sieve of Eratosthenes

Anonymous

89701 = 271 × 331

Anonymous

Probably the best way is to use Fermat's method, combined with some number sense. You will get a LOT of practice with your long division. You need to be able to test if a number is a perfect square, so you need to be able to extract square roots by hand. You avoid some work by recognizing that in base ten a perfect square must have the last two digits either be a perfect square, or a multiple of 20 plus a perfect square under 20. Another way to summarize that is that the last 2 digits must be 00, 25, e1, e4, o6 or o9; where e is an even digit and o is an odd digit. Fermat's idea is if a number has a factorization N = pq, with p>q>1, then you can let a=(p+q)/2 and b=(p-q)/2 so that a+b=p and a-b=q. (Try it.) Then N = (a+b)(a-b) = a^2 - b^2. If N is odd, then both p and q are odd, and then both a and b are integers. So, if an odd N can be factored, there must be integers a,b such that a^2 - N^2 = b^2. The smallest such pair will give the two divisors of N closest to sqrt(N) as (a-b) and (a+b). Saving myself from as many long divisions as possible, I note that your number N=89,701 is very close to 300^2 - 90,000, so that 299^2 = (300-1)^2 = 90,000 - 600 + 1 = 89,401. So 300^2 is the smallest square greater than N. Calculate a^2 - N = 90,000 - 89,701 = 299 and that's obviously not square. Next, 301^2 = 300^2 + 2*300 + 1 = 90,601 increases both a^2 and a^2-N by 601. You can build a table just by adding and subtracting: . a . . . . a^2 . . a^2 - N . . 300 . 90,000 . . . 299 . . . starting point, nonsquare 301 . 90,601 . . . 900 . . . add 601 (omigosh, hit one almost right away) 302 . 91,204 . . 1,503 . . . add 603 --- if we needed to go on 04 can end a square to do take square root 303 . 91,809 . . 2,109 . . . add 605 --- ditto, this would have needed a square root since 09 can end a square Those last two lines weren't needed The second line shows that: 301^2 -89,701 = 900 = 30^2 89,701 = 301^2 - 30^2 = (301 + 30)(301 - 30) = 331 * 271