Problem 81 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If the integers \(m\) and \(n\) are chosen at random between 1 and 100 then the probability that a number of the form \(7^{\mathrm{m}}+7^{\mathrm{a}}\) 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

It's not possible, an error in setup. Reticulate probabilities corrected.

Step 1: Understand the problem

We need to find the probability that a number of the form \(7^m + 7^n\) is divisible by 5, where \(m\) and \(n\) are integers between 1 and 100.

Step 2: Determine modulus pattern of powers of 7

Calculate the first few powers of 7 modulo 5 to identify a pattern: \(7^1 \equiv 2 \mod 5\), \(7^2 \equiv 4 \mod 5\), \(7^3 \equiv 3 \mod 5\), \(7^4 \equiv 1 \mod 5\). Notice that this pattern repeats every 4 cycles.

Step 3: Use modulus pattern to define conditions

Given the repeating cycle of 4, \(7^m + 7^n\) is divisible by 5 if \(7^m \equiv -7^n \mod 5\). Using \(7^m \equiv 2, 4, 3, 1 \mod 5\), find the conditions for their sum to equal a multiple of 5.

Step 4: Find valid m, n pair combinations

List combinations: if \(7^m \equiv 2 \mod 5\), \(7^n \equiv 3 \mod 5\) (or vice versa); if \(7^m \equiv 4 \mod 5\), \(7^n \equiv 1 \mod 5\) (or vice versa). Thus, there are 2 possible valid pairs for every 4 possible pairs.

Step 5: Calculate total possibilities

Both \(m\) and \(n\) take values from 1 to 100, leading to \(100 \times 100 = 10,000\) possible combinations.

Step 6: Calculate probability of divisibility by 5

Given the cycle has 4 possible powers (from step 2), within these, there are 2 successful combinations for divisibility. So, for each cycle of 4, there are 2 successful outcomes. Probability = \(\frac{2}{4} = \frac{1}{2}\) for each cycle.

Step 7: Adjust probability for full range

Considering there are 25 complete cycles in the range of 1 to 100, the probability is consistent across them, resulting as \(\frac{1}{2}\) across all cycles. Thus, overall probability = \(\frac{2}{4} = \frac{1}{2}\) = \(\frac{1}{2}\).

Key Concepts

Modular Arithmetic

In modular arithmetic, numbers wrap around when they reach a certain value known as the modulus. Think of a clock, which after reaching 12, starts over at 1. Here, if we talk about numbers modulo 5, any number gets reduced to a remainder between 0 and 4. For instance, 7 modulo 5 is 2 because when you divide 7 by 5, it leaves a remainder of 2.

This concept helps in simplifying problems, especially those involving powers. Instead of calculating large powers, you use the repeating remainder patterns. In the problem, we calculated the first few powers of 7 modulo 5 and found that they repeat every 4 cycles: 2, 4, 3, 1. Knowing this makes it much easier to predict results for any power of 7 using this cycle.

Number Theory

Number theory is a branch of mathematics that deals with the properties and relationships of numbers, especially integers. It's called the "queen of mathematics" for its foundational role in other areas.

In problems like the one given, number theory provides tools to tackle divisibility, prime numbers, and modular arithmetic. The concepts of patterns and cycles are vital, as they offer a systematic way to simplify and solve even complex arithmetic problems.

Using number theory, we can find patterns in sequences (like powers of 7), making it easier to determine characteristics like divisibility by a given number, as shown in the problem where we identified conditions for divisibility by 5.

Integers

Integers are whole numbers that can be positive, negative, or zero. They form the backbone of number theory and are fundamental in mathematics.

In our exercise, integers are chosen between 1 and 100 for values of both \(m\) and \(n\). This range results in 10,000 possible combinations for \(7^m + 7^n\). Understanding how these integers interact with modular arithmetic allows us to determine when such combinations satisfy certain properties, such as divisibility.

When dealing with integers in modular calculations, their straightforward nature often simplifies the process by dealing with only whole numbers.

Power Cycles

Power cycles refer to the repeating pattern of results when raising a number to successive powers under a modulus. In the context of this problem, noticing that powers of 7 repeat every 4 is a crucial observation.

Every time you raise 7 to successive powers, the sequence of remainders modulo 5 remains the same: 2, 4, 3, 1. These cycles allow us to predict results without direct computation of large powers. It's like having a shortcut to solving problems with exponentially growing numbers.

In practical terms, this means fewer calculations and an easy way to see conditions, like determining successful combinations \((m, n)\) for which \(7^m + 7^n\) is divisible by 5. As long as you know the start of the cycle, you can determine any future result.