There are five cards lying on the table in one row. Five numbers from

Question

There are five cards lying on the table in one row. Five numbers from among 1 to 100 have to be written on them, one number per card, such that the difference between the numbers on any two adjacent cards is not divisible by 4. The remainder when each of the 5 numbers is divided by 4 is written down on another card (the 6th card) in order. How many sequences can be written down on the 6th card?

A. 2^10

B. 2^10 x \(3^3\)

C. \(4 × 3^4\)

D. \(4^2 × 3^3\)

E. \(10^2\)

A. 2^10

B. 2^10 x  3^3

C.  4 × 3^4

D.  4^2 × 3^3

E.  10^2

nick1816 · Answer

The remainder when first number is divided by 4 can be 0,1,2 or 3
Since the difference between the numbers on any two adjacent cards is not divisible by 4, the remainder when second number is divided by 4 can be any except the remainder of the first number when divided by 4. Same is the case for other 3 numbers.

Suppose the sequence written on 6th card is 'abcde'.

'a' can take 4 values. Rest of them will take 3 values.

Number of sequences can be written = 4*3*3*3*3

ARIEN3228 · Answer

Here is the solution:

The remainder on the first card can be 0,1,2 or 3 i.e 4 possibilities.
The remainder of the number on the next card when divided by 4 can have 3 possible values (except the one occurred earlier).

For each value on the card the remainder can have 3 possible values.

Option: C

The total number of possible sequences is: 4*3^4

KarishmaB · Answer

Any positive integer will be of one of the four forms: 
4a / 4a+1 / 4a+2 / 4a+3
e.g. 6 = 4a + 2, 19 = 4b + 3, 21 = 4c + 1 etc

If difference between 2 numbers is not divisible by 4, it means the two numbers are not of the same form e.g. both cannot be of the form 4b + 3 such as 19 and 27. Hence the remainders they will leave on division by 4 cannot be the same.

This means that the sequence of 5 remainders needs to be such that no 2 adjacent remainders are the same. The first remainder can take one of 4 values (0/1/2/3). The second remainder can take one of 3 values (it cannot take the same value as the first remainder). Similarly, the third remainder can take one of 3 values (it cannot take the same value as the second remainder). The fourth remainder can take one of 3 values (it cannot take the same value as the third remainder). 
Then number of different sequences = 4 * 3 * 3 * 3 * 3  = 4 * 3^4

Hence examples of sequences of remainders are   or   etc

Answer (C)

bumpbot · Answer

Automated notice from GMAT Club BumpBot: 

A member just gave Kudos to this thread, showing it’s still useful. I’ve bumped it to the top so more people can benefit. Feel free to add your own questions or solutions.

 This post was generated automatically.

There are five cards lying on the table in one row. Five numbers from

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Problem Solving (PS) Questions

There are five cards lying on the table in one row. Five numbers from

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards