GMAT Data Sufficiency - What is the Remainder when 333^222 is Divided by 7?

Question: What is the remainder when 333^222 is divided by 7?

1. 3
2. 2
3. 5
4. 7
5. 1

‘What is the remainder when 333^222 is divided by 7?’ – is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book "GMAT Quantitative Review". The GMAT Quant section consists of a total of 31 questions. GMAT Data Sufficiency questions consist of a problem statement followed by two factual statements. GMAT data sufficiency comprises 15 questions which are two-fifths of the total 31 GMAT quant questions.

Solution and Explanation:

Approach Solution 1:

The given question requires 333^222 to be divided by 7 and to find the remainder.

333^222 = (329+4)^222 = (7∗47+4)

Now if we expand this, all terms but the last one will have 7*47 as a multiple and thus will be divisible by 7. The last term will be 4^222 = 2^444
Hence, it is important to find the remainder when 2^444 is divided by 7

2^1 divided by 7 yields remainder of 2;
2^2 divided by 7 yields remainder of 4;
2^3 divided by 7 yields remainder of 1;
2^4 divided by 7 yields remainder of 2;
2^5 divided by 7 yields remainder of 4;
2^6 divided by 7 yields remainder of 1;

The remainder repeats in blocks of three: {2-4-1}. So, the remainder of 2^444 divided by 7 would be the same as 2^3 divided by 7 (444 is a multiple of 3). 2^3 divided by 7 yields remainder of 1.

Correct Answer: E

Approach Solution 2:

Given in the problem: (333)^222
Remainder comes (333/7) = 4
This implies (4) ^222
This implies (16) ^111
Remainder comes (16/7) = 2
(2)^111 = 2^100 * 2^11
Now let us observe the Remainder comes(2^10)/7
2^10 = 1024

This implies that Remainder comes (1024/7) = 2
Remainder(2^100 * 2^11 ) be 7 = Remainder((2^10)^10 * 2^11) by 7

This implies that Remainder comes((2)^10 * 2^10 * 2) by 7
Remainder( 2* 2 *2 ) by 7
That implies 8/7
That implies 1

Correct Answer: E

Approach Solution 3:

According to Ferment’s Theorem:

Given, (a)^x/p

The remainder of (a)^x/p will be 1

so the cases may apply that

1. ‘a’ and ‘p’ both are coprime numbers
2. ‘p’ is a prime number
3. x=Ø(p)*k

here, k is constant;

Ø(p) =(p-1)

this implies x=(p-1)k

this suggests (a)^p-1)k/p has remainder 1(satisfying I and 2 conditions)

Correct Answer: E

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