Question

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Answer 1

Răspuns:

a = 3¹ + 3²+...+3⁴⁸

a = 3·(1+3¹ + 3²+...+3⁴⁷)

a = 3·(1+3¹ + 3²+...+3⁴⁷+3⁴⁸ - 3⁴⁸)

a = 3·(1+a - 3⁴⁸)

a = 3+3a - 3·3⁴⁸

2a =3·3⁴⁸ -3

2a =3·(3⁴⁸ -1)

2a =3·[(3²⁴)² -1²]

2a =3·(3²⁴ -1)(3²⁴ +1)

2a =3·(3¹² -1)(3¹²+1)(3²⁴+1)

2a =3·(3⁶ -1)(3⁶ +1)(3¹²+1)(3²⁴+1)

2a =3·(3³ -1)(3³+1)(3⁶+1)(3¹² +1)(3²⁴+1)

2a =3·(27 -1)(3³+1)(3⁶+1)(3¹²+1)(3²⁴+1)

2a =3·26·(3³+1)(3⁶+1)(3¹²+1)(3²⁴+1)

2a =3·2·13·(3³+1)(3⁶+1)(3¹²+1)(3²⁴+1)

a =3·13·(3³+1)(3⁶+1)(3¹²+1)(3²⁴+1)

a =13·k, k = 3·(3³+1)(3⁶+1)(3¹²+1)(3²⁴+1)

⇒ 13 | a