Factoring a large number

Which of the following is the correct prime factorization of 414863?

577 × 709 577 \times 709 577 × 719 577 \times 719 587 × 709 587 \times 709 587 × 719 587 \times 719

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6 solutions

Nikola Djuric
Feb 19, 2016

(600-10 m-3) (700+10*n+9)=420000-5137

so 420000+6000n+5400-7000m-100mn-90m-30n-2100-21=-5137

so 1000(6n-7m)+3300=-5110+30(n+3m)

so right side is divided by 100,so 30(n+3m)=10 (mod 100)

n+3m=7,so it is now easy n=1, m=2

so our product is (600-23)*(700+19)=577x719

OR check 77*09=693 so the last 2 digits are 93

or 87*09+=783 so the last two digits are 83

87*19=1740-87=1653,so the last two digits are 53

77*19=1540-77=1500-37=1463,so the last two digits are 63, so answer is 577x719

Nikola Djuric - 5 years, 3 months ago
Ivan Koswara
Jan 12, 2016

Just compute the value given from each choice. Since all of them are different, we can just pick the one that gives the correct result; we don't even need to verify that the individual factors are primes.

Ashley Slappey
Dec 7, 2020

solve the problem using prime factorization.

Yeah, like why didn't everyone here think of that...

Nerdy Nachos - 5 months, 1 week ago
Bip 901
Feb 9, 2019

A brute-force way, but x2 more efficient: I divided by 577 (the first divisor in the answers), got an integer (719), answered! If I hadn't gotten an integer I would have checked the next divisor, 587 (no need to check the 2nd answer).

Mohamed Wafik
Feb 6, 2016

The last digit of all choices is 3 but the second will be (6+3+(0 for first, 7 for second, 9 for third or (9+7->6) for forth choice) which can only give 6 for second digit using the second choice

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