Uzasadnij, że liczba 5 (do potęgi 16) - 1 jest podzielna przez 24
5^16 - 1 = ( 5^8)^2 - 1^2 = ( 5^8 - 1)*( 5^8 + 1) = [ (5^4)^2 - 1)]*( 5^8 + 1) =
= ( 5^4 - 1)*( 5^4 + 1)*( 5^8 + 1) = [ ( 5^2)^2 - 1]*( 5^4 + 1)*(5^8 + 1) =
= ( 5^2 - 1)*(5^2 + 1)*(5^4 + 10*(5^8 + 1) =
= ( 5 - 1)*( 5 + 1)*(5^2 + 1)*(5^4 + 10*(5^8 + 1) =
= 4*6 *( 5^2 + 1)*(5^4 + 1)*(5^8 + 1) =
= 24*(5^2 + 1)*( 5^4 + 1)*(5^8 + 1) - czyli liczba podzielna przez 24
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ckd.
Korzystamy kilka razy z wzoru
a^2 - b^2 = ( a - b)*( a + b)
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5^16 - 1 = ( 5^8)^2 - 1^2 = ( 5^8 - 1)*( 5^8 + 1) = [ (5^4)^2 - 1)]*( 5^8 + 1) =
= ( 5^4 - 1)*( 5^4 + 1)*( 5^8 + 1) = [ ( 5^2)^2 - 1]*( 5^4 + 1)*(5^8 + 1) =
= ( 5^2 - 1)*(5^2 + 1)*(5^4 + 10*(5^8 + 1) =
= ( 5 - 1)*( 5 + 1)*(5^2 + 1)*(5^4 + 10*(5^8 + 1) =
= 4*6 *( 5^2 + 1)*(5^4 + 1)*(5^8 + 1) =
= 24*(5^2 + 1)*( 5^4 + 1)*(5^8 + 1) - czyli liczba podzielna przez 24
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ckd.
Korzystamy kilka razy z wzoru
a^2 - b^2 = ( a - b)*( a + b)
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