Question

FIND THE REMAINDER WITH DETAILED SOLUTION

Answer 1

Answer:

To find: Remainder when $17^{18^{19^{20^{...{∞} } } } }$

• We use congruency modulo to find solutions to these types of questions.
• It is specially used to find remainders in an easy way.
• A ≡ B (mod C) This says that A is congruent to B modulo C.

• ≡ is the symbol for congruence, which means the values A and B are in the same equivalence class.

Note: A is congruent to B modulo C

           

Here, 17 ≡ 1 (mod 8)

⇒ We know that if A ≡ B (mod C) then Aⁿ ≡ Bⁿ (mod C)

We can apply that theorem here

⇒ 17ⁿ ≡ 1ⁿ (mod 8)

If we take 'n' as $18^{19^{20^{...{∞} } } } }$

⇒ $17^{18^{19^{20^{...{∞} } } } }$≡ $1^{18^{19^{20^{...{∞} } } } }$(mod 8)

Any this to the power of 1 is 1.

So, $17^{18^{19^{20^{...{∞} } } } }$≡ 1 (mod 8)

∴ 1 is the remainder to your question.

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