Question

If N = 314 + 313 – 12, then what is the largest prime factor of N?

Answer 1

Given : N = 3¹⁴ + 3¹³ - 12

To Find :  largest prime factor of N

सबसे बड़ा अभाज्य गुणनखण्ड

Solution:

N = 3¹⁴ + 3¹³ - 12

= 3¹³( 3 + 1) - 12

=  3¹³ (4) - 3( 4)

= 4 ( 3¹³ - 3)

=  4 . 3 ( 3¹² - 1)

= 4 . 3 ( (3⁶)² - 1²)

using a² - b² = (a + b)(a - b)

=  4 . 3 ( 3⁶ + 1)(3⁶ - 1)

= 4 . 3 ( 3⁶ + 1)(3³ + 1)(3³ - 1)

= 4 . 3  ( 3⁶ + 1)(28)(26)

= 4 . 3 (2.2.7)(2.13) ( 3⁶ + 1)

= 4 . 3 (2.2.7)(2.13) ( (3²)³ + 1³)

a³ + b³ = ( a + b)(a² +b² - ab)

= 4 . 3 (2.2.7)(2.13) ( 3² + 1)(3⁴ + 1 - 3²)

= 4 . 3 (2.2.7)(2.13) (10)(81 + 1 - 9)

= 4 . 3 (2.2.7)(2.13) (2.5)(73)

= 2.2 . 3 (2.2.7)(2.13) (2.5)(73)

= 2⁶.3..5.7.13.73

3¹⁴ + 3¹³ - 12 = 2⁶.3..5.7.13.73

largest prime factor  = 73

सबसे बड़ा अभाज्य गुणनखण्ड = 73

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