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What is the remainder when \(5555^{2222}+2222^{5555}\) is divided by 7?
--> \((7m+4)^{2222}+(7n+3)^{2222}= ... + 4^{2222}+ ...+ 3^{5555}\)

\(4^{2222}= (7-3)^{2222}= ...+3^{2222}\)
-------------------------------------------

-->\(3^{2222} + 3^{5555}= 3^{2222}*(1+ 3^{3333})= 3^{2222}*(1+ 27^{1111})= 3^{2222}*(1+ (28-1)^{1111})\)

--> \(3^{2222}*(1+ (7x^{1111}+ ...-(1)^{1111})= 3^{2222}*(1+ (7x)^{1111}+ ...-1)= 3^{2222}*(7x)^{1111}\)

\(3^{2222}*(7x)^{1111}\) can be divided by 7. The remainder will be 0

The answer choice A is Correct
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Bunuel
What is the remainder when \(5555^{2222} + 2222^{5555}\) is divided by 7?

A. 0
B. 1
C. 3
D. 4
E. 6



Concept : Remainder of when \(5555^{2222}\) is divided by 7 = Remainder when \(4^{2222}\) where Remainder [5555/7] = 4

Concept : Remainder of when \(2222^{5555}\) is divided by 7 = Remainder when \(3^{5555}\) where Remainder [2222/7] = 3


Remainder \(5555^{2222} + 2222^{5555}\) divided by 7 = Remainder \(4^{2222} + 3^{5555}\) divided by 7 = 0 {because 4+3=7}
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Bunuel
What is the remainder when \(5555^{2222} + 2222^{5555}\) is divided by 7?

A. 0
B. 1
C. 3
D. 4
E. 6

Are You Up For the Challenge: 700 Level Questions

Asked: What is the remainder when \(5555^{2222} + 2222^{5555}\) is divided by 7?

Remainder when 5555^{2222} + 2222^{5555} is divided by 7
= Remainder when 4^{2222} + 3^{5555} is divided by 7
= Remainder when 4^2 + (-1)^{1851}*3^2 is divided by 7 since 4^3 = 64 = 63 + 1 and 3^3 = 27 = 28 -1
= Remainder when 16 - 9 = 7 is divided by 7
= 0


IMO A
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step 1: 5555/7
55/7 = 6 rem
65/7 = 2 rem
25/7 = 4rem
so 4^2 =16
16/7 = 2 rem-------------- (i)
Step 2: 2222/7
22/7 = 1 rem
12/7 = 5rem
52/7 =3 rem
now 3^5 = 9*9*3
again 9/7 = 2 rem
9/7 = 2 rem
2*2*3/7 = 5 rem----------- (ii)
add (i+ii)
(2+5)/7 = 0 remainder that is the ans

if anyone have any query plz ask
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Why can't we just estimate the units digit and figure out the remainder based on the units digit?
Would be quicker no? Solved it very quickly and got the correct answer. Is it the approach correct or did I just get lucky?
Bunuel
What is the remainder when \(5555^{2222} + 2222^{5555}\) is divided by 7?

A. 0
B. 1
C. 3
D. 4
E. 6

Are You Up For the Challenge: 700 Level Questions
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Puja01
Why can't we just estimate the units digit and figure out the remainder based on the units digit?
Would be quicker no? Solved it very quickly and got the correct answer. Is it the approach correct or did I just get lucky?
Bunuel
What is the remainder when \(5555^{2222} + 2222^{5555}\) is divided by 7?

A. 0
B. 1
C. 3
D. 4
E. 6

Are You Up For the Challenge: 700 Level Questions

The remainder when dividing by 7 is not determined by the units digit. For example, 11 divided by 7 gives a remainder of 4, but 21 divided by 7 gives 0. So using just the last digit doesn’t work here.
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