Problem 5 - A Complete Resource Book in Mathematics for JEE Main

Question

If the integers \(m\) and \(n\) are chosen at random between 1 and 100 then the probability that a number of the form \(7^{w}+7^{x}\) is divisible by 5 is (A) \(\frac{1}{5}\) (B) \(\frac{1}{7}\) (C) \(\frac{1}{4}\) (D) \(\frac{1}{49}\)

Step-by-Step Solution

Answer

(C) \(\frac{1}{4}\)

Step 1: Understanding Powers of 7 Modulo 5

We first need to find the remainder when powers of 7 are divided by 5. This is represented as \(7^k \mod 5\). Let's compute for several values of \(k\).\- \(7^1 \equiv 2 \mod 5\)\- \(7^2 \equiv 4 \mod 5\)\- \(7^3 \equiv 3 \mod 5\)\- \(7^4 \equiv 1 \mod 5\)\We find a repeating cycle every 4 terms: \(2, 4, 3, 1\).

Step 2: Determine all possible combinations for divisibility by 5

For \(7^w + 7^x\) to be divisible by 5, we need the sum of \(7^w \mod 5\) and \(7^x \mod 5\) to be \(0 \mod 5\). Check combinations:\- If \(7^w \equiv 2 \): \(7^x \equiv 3\)\- If \(7^w \equiv 3 \): \(7^x \equiv 2\)\- If \(7^w \equiv 1 \): \(7^x \equiv 4\)\- If \(7^w \equiv 4 \): \(7^x \equiv 1\)

Step 3: Analyze the Probability

Given the repeating cycle of 4 for \(7^k\mod 5\), each integer \(m\) or \(n\) corresponds to one of these remainders: \(2, 4, 3,\) or \(1\), with each outcome equally probable.\- There are 4 pairs of residues in a cycle that satisfy the divisibility condition per cycle of 4 and there are \(4 \times 4 = 16\) possible \((w,x)\) residues combinations as each has 4 options.\- Probability = \(\frac{Number\, of\, favorable\, outcomes}{Total\, outcomes} = \frac{4}{16} = \frac{1}{4}\)

Step 4: Conclusion

The solution shows that when integers \(m\) and \(n\) are selected, the probability that \(7^w + 7^x\) is divisible by 5 is \(\frac{1}{4}\). Therefore, the answer is (C) \(\frac{1}{4}\).

Key Concepts

Modular Arithmetic

At the heart of this problem lies the concept of modular arithmetic, an extremely handy tool in mathematics used to find remainders after division. Imagine you have a clock. The numbers go from 1 to 12 and then loop back. This is similar to how modular arithmetic works, where numbers wrap around after reaching a certain value (known as the modulus).
In the context of this exercise, we are calculating powers of 7 modulo 5. This means we are interested in the remainder when each power of 7 is divided by 5. For example, when 7 is raised to the power of 1, it leaves a remainder of 2 when divided by 5: thus, we write this as \(7^1 \equiv 2 \mod 5\). Each subsequent power of 7 follows this pattern and reveals a repeating cycle: 2, 4, 3, 1 every four numbers. Understanding this cyclical nature is crucial for solving problems that involve finding probabilities related to these powers.

• The cycle, in this case, results from repeated division by 5.
• Remainders form a pattern that repeats every four steps: 2, 4, 3, and 1.

This repetition helps delve deeper into the problem and directly impacts calculating probabilities, setting the stage for the next concept in focus.

Divisibility Rules

Divisibility rules are simple shortcuts that help determine if a number can be divided by another number without leaving a remainder. In our exercise, we explore if the number formed by adding two powers of 7 is divisible by 5.
To be divisible by 5, the sum of these powers, represented here as \(7^w + 7^x\), must result in a multiple of 5. In modular terms, this means that \((7^w + 7^x) \equiv 0 \mod 5\). This can occur when the residues (remainders) of \(7^w \mod 5\) and \(7^x \mod 5\) sum to 5, ultimately producing no remainder when divided.

• Possible pairings producing a sum divisible by 5 are: (2, 3) and (3, 2); and (4, 1) and (1, 4).

Each of these pairings gives us insights into how many combinations meet our criteria. The understanding of these basic rules can greatly simplify evaluations of divisibility, especially in mathematical probability puzzles.

Integer Sequences

Integer sequences form an important part of mathematics, as they are simply ordered lists of numbers arranged according to some rule or pattern. In the context of our problem, the sequence arises from the remainders calculated using powers of 7.
The sequence \(2, 4, 3, 1\) repeats every four numbers, providing a cycle that can predict the result of any arbitrary power of 7 when divided by 5. Knowing that this sequence is predictable and repeats, we can calculate the probability of each residue occurring in a vast range of numbers. This knowledge guides us in identifying which combinations of these numbers add up to a desired sum divisible by 5.

• When dealing with powers of numbers, predicting patterns is key.
• This cycle of remainders is a typical example of integer sequences.

Understanding these cyclic sequences changes how we approach and solve problems, as it reduces complexity and highlights patterns to take advantage. This process enables a methodical approach in solving the exercise efficiently.