show that 16^99 - 1 = 0 ( mod 437 )
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The proof of 1 - 1 = 0 (mod 437) is given below:
- To solve this problem, we will be using the following concepts:
1) a = b (mod n) ⇒ = (mod n)
2) If a = b (mod p) and a = b (mod q) where GCD (p , q) = 1, then
a = b(mod pq)
- We know that 437=19×23 and GCD (19 , 23) = 1
Now,
(−3)³ = −8(mod 19)
⇒ = 1 (mod 19)
⇒ 1 = 1 (mod 19)
⇒ 1 = 1 (mod 19) ....................(I)
- Also,
(−7)² = 3 ( mod 23)
⇒ ( ) = 9 (mod 23)
⇒ () = 12 (mod 23)
⇒ (1 ) = 1 (mod 23)
⇒ 1 = 1 (mod 23) ....................(II)
- From (I) and (II)
1 = 1 (mod 19x23) = 1 (mod 437)
⇒ 1 - 1 = 0 (mod 437)
Hence Proved.
#SPJ3
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