Five cards are placed in a row, and each card shows a number from 1 to 100, with one number per card, such that the difference between the numbers on any two adjacent cards is not divisible by 4. When each number is divided by 4, its remainder (0, 1, 2 or 3) is written, in order, on a sixth card. How many different remainder sequences can appear on this sixth card?

Introduction / Context:
This question uses modular arithmetic and sequence counting. Instead of focusing on the original numbers between 1 and 100, the condition only depends on their remainders when divided by 4. The key idea is to recognize that the adjacency condition translates into a simple rule on these remainders, which then allows us to count all valid remainder sequences of length 5.

Given Data / Assumptions:

• Five numbers are chosen from 1 to 100, one on each of 5 cards.
• Difference between numbers on adjacent cards is not divisible by 4.
• For each number, we record its remainder when divided by 4: this remainder is 0, 1, 2 or 3.
• The sequence of 5 remainders is written on a sixth card.
• We must count how many length 5 sequences of remainders are possible under the adjacency rule.

Concept / Approach:
If the difference between two numbers is divisible by 4, then their remainders modulo 4 are equal. Therefore, the condition “difference not divisible by 4” means that no two adjacent numbers can have the same remainder modulo 4. So we are counting sequences of length 5 over the set {0, 1, 2, 3} such that adjacent elements are never equal.

Step-by-Step Solution:
Position 1 (first card): any of the 4 remainders {0, 1, 2, 3} can appear. So there are 4 choices.Position 2: it cannot equal the remainder in position 1, so there are 3 possible remainders.Position 3: it cannot equal the remainder in position 2, so again 3 choices.Position 4: cannot equal the remainder in position 3, so 3 choices.Position 5: cannot equal the remainder in position 4, so 3 choices.Total sequences = 4 * 3 * 3 * 3 * 3 = 4 * 3^4.

Verification / Alternative check:
You can expand 3^4 as 81 and then 4 * 81 = 324.Thus there are 324 valid remainder sequences; the option given in functional form is 4 x 3^4.

Why Other Options Are Wrong:
3^4 accounts for choices assuming the first position has only 3 options, which is incorrect.4^3 severely underestimates the number of sequences and does not match the positional logic.3 x 4^3 treats the constraints incorrectly and overcounts some invalid patterns.

Common Pitfalls:
Misreading the condition and allowing equal remainders in adjacent positions.Trying to work directly with the numbers from 1 to 100 instead of reducing the problem to remainders modulo 4.Forgetting that the first position has 4 options while subsequent positions are constrained to 3 each.

Final Answer:
The number of possible remainder sequences is 4 x 3^4.