Question

The greates positive integer k for which (49^k)+1 is a factor of sum 49^125+49^124...........+1 is

Answer 1

k = 63  for which (49^k)+1 is a factor of sum  49¹²⁵ + 49¹²⁴  + ......................................................+ 1

Step-by-step explanation:

49¹²⁵ + 49¹²⁴  + ......................................................+ 1

= 1  + 49  + ...................................................+ 49¹²⁴ + 49¹²⁵

a = 1

r = 49

n = 126

Sum =  a( rⁿ - 1)/(r - 1)

= 1(49¹²⁶ - 1)/(49 - 1)

= (49¹²⁶ - 1)/48

= ((49⁶³)² - 1²)/48

using a² - b² = (a + b)(a - b)

= (49⁶³ + 1)(49⁶³ - 1)/48

49⁶³ + 1 is a factor

Hence  k = 63

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