Question: What is the remainder when \(47^{203}\) is divided by 7?

1. 1
2. 2
3. 3
4. 4
5. 5

Answer:
Solution with Explanation:
Approach Solution (1):

We can find the pattern by finding remainders when successive powers are divided.

\(47^1\) leaves 5, \(47^2\) leaves 4 and so on.. pattern is 5, 4, 6, 2, 3, 1, 5, 4… Thus, the pattern repeats after every 6th number.
So convert 203 in 6k form… 203 = 6k + 5
So remainder will be 5th in the pattern 5, 4, 6, 2, 3 1, hence 3

Correct Option: C

Approach Solution (2):

Here, \(47^{203}=(49-2)^{203}\)

All terms on expansion except\((-2)^{203}\)will be divisible by 7.

\((-2)^{203}=(-2)^{202}*(-2)=(2^3)^\frac{201}{3}*(2)*(-2)\)\(=(8)^{67}*2*(-2)=(7+1)^{67}*(2)*(-2)=1^{67}*(-4)=-4\)

Remainders cannot be negative, so remainder = 7 – 4 = 3

Correct Option: C

“What is the remainder when \(47^{203}\) is divided by 7?”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book “GMAT Official Guide Quantitative Review”. To solve GMAT Problem Solving questions a student must have knowledge about a good amount of qualitative skills. The GMAT Quant topic in the problem-solving part requires calculative mathematical problems that should be solved with proper mathematical knowledge.

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