Clever Factoring

Number Theory Level 3

N 1 N_1 and N 2 N_2 are two 4-digit numbers whose product is 4 8 + 6 8 + 9 8 4^8+6^8+9^8

Find N 1 N_1 and N 2 N_2 , express your answer as N 1 + N 2 N_1+N_2 .


The answer is 13634.

Bloons Qoth by Bloons Qoth , 21, USA

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1 solution

Tom Engelsman
Oct 17, 2016

Observe:

4^8 + 6^8 + 9^8 = 2^16 + (2^8)(3^8) + 3^16;

= 2^16 + 2*(2^8 * 3^8) + 3^16 - (2^8 * 3^8);

= (2^8 + 3^8)^2 - 6^8;

= [(2^8 + 3^8) + 6^4][(2^8 + 3^8) - 6^4];

= (N1)(N2).

Thus, N1 + N2 = 2*(2^8 + 3^8) = 13,634.

2 Helpful 0 Interesting 0 Brilliant 0 Confused

Playing devil's advocate, you should explain why there aren't other possible pairs of 4-digit numbers that satisfy the conditions of the question.

Calvin Lin Staff - 4 years, 7 months ago

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