What is the Remainder When 43^86 is Divided by 5? GMAT Problem Solving

Question: What is the remainder when 43^86 is divided by 5?

Correct Answer: E

Solution and Explanation
Approach Solution 1:

It is asked in the question to find out the remainder when \(43^{86}\)is divided by 5.
This question is from the binomial theorem.
It is advised to read binomial theorem before solving this problem.
We can write \(43^{86}\)as \((40+3)^{86}\)
\((40+3)^{86}\)
In the expansion of this equation all the terms will contain the term 40 except the last term.
The last term will be \(3^{86}\).
Since 40 is divisible by 5, only last term will be the deciding factor for the remainder.
\(3^{86}\)
It can be re written as,
\((3^2)^{43}\) = \(9^{43}\)= (10 - 1\()^{43}\)
The expansion of this equation will contain 10 in all the terms except the last term.
The last term will be (-1\()^{43}\) = -1
As the last term is negative the remainder will be 5 - 1 = 4

Approach Solution 2:
It is asked in the question to find out the remainder when \(43^{86}\)is divided by 5.
This question is from the binomial theorem.
It is advised to read binomial theorem before solving this problem.
We can write \(43^{86}\)as \((40+3)^{86}\)
\((40+3)^{86}\)
In the expansion of this equation all the terms will contain the term 40 except the last term.
The last term will be \(3^{86}\).
Now only the units digit is required to know the remainder.
Now we’ll look for the patterns in the exponents of 3
\(3^1\) = 3 (the unit’s digit is only written here)
\(3^2\)= 9
\(3^3\)= 7
\(3^4\)= 1
\(3^5\)= 3
\(3^6\)= 9
…

There is a pattern of 3 - 9 - 7 -1 ..3 - 9 - 7 -1
Now when we see the number \(3^{86}\), when 86 is divided by 4 it will give 2 as remainder
Therefore the units term in \(3^{86}\)will be equal to unit term in \(3^2\)= 9
The remainder when this is divided by 5 = 9/5 = 4

Approach Solution 3:
All the terms except 3^86 will have 40 as a multiplier and hence would be divisible by 5

3^1÷5 remainder 3
3^2÷5 remainder 4
3^3÷5 remainder 2
3^4÷5 remainder 1
3^5÷5 remainder 3
and then 4, 2 and 1 will follow repetitively

repeat happens after every 4

86÷4 remainder 2

so 3^2 case would be applicable and hence remainder 4

“What is the remainder when 43^86 is divided by 5?”- s a topic of the GMAT Quantitative reasoning section of GMAT. To solve GMAT Problem Solving questions a student must have knowledge about a good amount of qualitative skills. GMAT Quant practice papers improve the mathematical knowledge of the candidates as it represents multiple sorts of quantitative problems.

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What is the Remainder When 43^86 is Divided by 5? GMAT Problem Solving

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