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Find the non-negative remainder when 4165 and 250 are divided by 7

Find the non-negative remainder when 4165 and 250 are divided by 7 Correct Answer 6 and 4 respectively

Concept:

Let n be fixed positive integer, then two integers a and b are congruent modulo n, denoted by a ≡ b(mod n) if n divides (a - b).i.e., (a - b) = kn for some k∈ Z, and if n doesn´t divide (a - b) then a and b are non-congruent modulo n.

Calculation:

Consider 4165 

we know that, 41 ≡ -1(mod 7)        ....(1)

rise the power to 64 on both the sides, we get

(41)64 ≡ (-1)64 (mod 7)

⇒ (41)64 ≡ 1 (mod 7)

multiply both the sides by 41, we have

⇒ (41)64.41 ≡ 1 . 41 (mod 7)

⇒ (41)65 ≡ 41 (mod 7)

⇒ (41)65 ≡ - 1 (mod 7) | using (1)

⇒ (41)65 ≡ 6 (mod 7) 

Thus the non-negative remainder when 4165 is divided by 7 is 6.

Similarly, consider  250

we know that 21 ≡ 2(mod 7) 

and 22 ≡ 4 (mod 7) 

and 23 ≡ 1 (mod 7) 

rise the power to 16 on both the sides, we get

(23)16 ≡ 116 (mod 7) 

⇒ 248 ≡ 1 (mod 7) 

multiply both the sides by 22, we have

⇒ 248. 22 ≡ 1 . 22 (mod 7)

⇒ 250 ≡ 4 (mod 7) 

Thus the non-negative remainder when 250 is divided by 7 is 4.

Hence, the correct answer is option 3)

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