A security system flashes a red light every time the number of login attempts reaches a power of 3. The control software keeps track of this and checks the value of 3125. To verify system synchronization, the technician needs to know what remainder this number leaves when divided by 7. What will be the remainder?
(a) 6
(b) 4
(c) 5
(d) 3

Question

A security system flashes a red light every time the number of login attempts reaches a power of 3. The control software keeps track of this and checks the value of 3125. To verify system synchronization, the technician needs to know what remainder this number leaves when divided by 7. What will be the remainder?
(a) 6
(b) 4
(c) 5
(d) 3

ONLiNE UPSC · Accepted Answer

We calculate, the remainder of 3125 when divided by 7.
Observe the pattern:
31 = 3 mod&thinsp;7 = 3
32 = 9 mod&thinsp;7 = 2
33 = 27 mod&thinsp;7 = 6
34 = 81 mod&thinsp;7 = 4
35 = 243 mod&thinsp;7 = 5
36 = 729 mod&thinsp;7 = 1
37 = 2187 mod 7 = 3
38 = 6561 mod 7 = 2
So, the powers of 3 modulo 7 repeat after every 6 terms.
Now, 125 mod&thinsp;6 = 5, so
3125 mod 7 = 35 mod 7 = 5

Hence, option c is correct.

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Remainder Theorem

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M Question 1	⏱ 0
A security system flashes a red light every time the number of login attempts reaches a power of 3. The control software keeps track of this and checks the value of 3125. To verify system synchronization, the technician needs to know what remainder this number leaves when divided by 7. What will be the remainder? (a) 6 (b) 4 (c) 5 (d) 3