Question

7^4k ≡_____(mod 100) 1)1 2)2 3)3 4)4​

Answer 1

Answer: 7^(4k) ≡ 1 (mod 100)

Step-by-step explanation:

Let's try to observe a pattern :-

7⁰ ≡ 1 (mod 100)  [since 7⁰ = 1]

7¹ ≡ 7 (mod 100)

7² ≡ 49 (mod 100)

7³ ≡ 343 (mod 100)

=> 7³ ≡ 43 (mod 100)

7⁴ ≡ 301 (mod 100)

=> 7⁴ ≡ 1 (mod 100) .

Note that since the last 2 digits of 7⁴ ends with 01 so it gives 1 (mod 100)

Also note that this table again continues the same way from 7⁵,7⁶,7⁷ and so on , as 7⁰ also gave 1 (mod 100) .

So 7⁵ ≡ 7 (mod 100)

7⁶ ≡ 49 (mod 100)

7⁷ ≡ 43 (mod 100)

7⁸ ≡ 1 (mod 100) again .

Noticing the pattern , we get that 7^(every multiple of 4) ≡ 1 (mod 100) .

Since every multiple of 4 can be represented as 4k , for some number k , we conclude 7^(4k) ≡ 1 (mod 100) .

Hence the correct answer is option 1 .

Hope this helps you .

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Answer 2

Answer:

1 is the answer is correct