Largest prime factor of an Exponential Expression
Find the largest prime factor of 4 9 + 9 4 .
The answer is 881.
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2 solutions
But how did you verified its a prime
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the root of 881 is about 29. No number below 29 divides it.
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Can you please show me the steps a little more clearly?
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@Saran Balachandar – which part?
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@Dev Sharma – How did you use the formula in this problem?
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@Saran Balachandar – Sophie germain identity
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@Dev Sharma – Yeah I understood the factoring but I didn't understand how you got 881 as the largest prime factor from the factored expression.
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@Saran Balachandar – Dev's dots are multiplications. Multiplying out the two parenthetical expressions you get (305)(881). 305 can be dismissed as it isn't prime. And as Dev said, since the square root of 881 is 29 and change, you just need to see if 881 is divisible by any of the primes less than 29 (and check 29 itself too - just for grins).
@Saran Balachandar – You can just find the value of the terms and you will see that 881 is the largest prime factor.
@Saran Balachandar – It's like factoring a number of the form a^2+b^2.
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@Kushagra Sahni – To factor of the form a^4 + 4b^4
Same Way but I didn't know that this is called the Sophie Germain Identity.
Did the exact same
4 9 + 9 4 → 2 1 8 + 3 8 = 2 6 2 1 4 4 + 6 5 6 1 = 2 6 8 7 0 5 = 5 × 6 1 × 8 8 1 .
Even, 5 , 6 1 , and 8 8 1 are all prime, so 8 8 1 is the largest prime factor of 4 9 + 9 4 .
Using Sophie Germain Identity,
9 4 + 4 . 4 8 = ( 9 2 + 2 . 1 6 2 − 2 . 9 . 1 6 ) ( 9 2 + 2 . 1 6 2 + 2 . 9 . 1 6
solving gives 881 as largest prime factor.