Problem 10 - Algebra for JEE ADVANCED

Question

Given that the divisors of \(n=3^{p} \cdot 5^{q} \cdot 7^{\prime}\) are of the form \(4 \lambda+1\), \(\lambda \geq 0\). Then a. \(p+r\) is always even b. \(p+q+r\) is always odd c. \(q\) can be any integer d. if \(p\) is odd then \(r\) is even

Step-by-Step Solution

Answer

True: a, c, d; False: b

Step 1: Understanding the Problem

We know that the possible divisors of the number \(n = 3^p \cdot 5^q \cdot 7^r\) need to satisfy the form \(4\lambda + 1\). This implies each divisor is congruent to 1 modulo 4.

Step 2: Analyzing the Congruences

Let's analyze what numbers \(3, 5,\) and \(7\) are congruent to when considered modulo 4.- \(3 \equiv 3 \pmod{4}\)- \(5 \equiv 1 \pmod{4}\)- \(7 \equiv 3 \pmod{4}\)Since the divisors are concurrent to \(1 \pmod{4}\), any power or combination must equal \(1 \pmod{4}\).

Step 3: Examining Individual Powers

To get a product of divisors of the form \(4k + 1\), we need:- \(3^p \equiv 1 \pmod{4}\) if \(p\) is even since \(3^2 \equiv 1 \pmod{4}\)- \(5^q \equiv 1 \pmod{4}\) is always true as \(5 \equiv 1 \pmod{4}\)- \(7^r \equiv 1 \pmod{4}\) if \(r\) is even since \(7^2 \equiv 1 \pmod{4}\)

Step 4: Evaluating Condition a and d

To satisfy conditions:- Since \(p\) must be even, \(p + r\) is always even (So, a is true).- If \(p\) is odd to make the product \(1 \pmod{4}\), \(r\) must be even (So, d is also true).

Step 5: Evaluating Condition c

The power \(q\) does not affect the form \(4\lambda+1\) because \(5^q \equiv 1 \pmod{4}\) irrespective of \(q\). Therefore, \(q\) can be any integer (c is true).

Step 6: Evaluating Condition b

For \(p+q+r\) to make the entire expression \(1 \pmod{4}\), we check the equality:- Since \(p\) and \(r\) must both be even, \(p + q + r\) need not necessarily be odd; therefore, b is false.

Key Concepts

Congruences Modulo

Congruences modulo is a concept in number theory that helps in understanding how numbers behave when divided by another number, known as the modulus. In this exercise, divisors of the number \( n = 3^p \cdot 5^q \cdot 7^r \) have to be of the form \( 4\lambda + 1 \), meaning they must be congruent to 1 modulo 4.
The idea is to classify integers into different "residue classes" based on their remainder when divided by 4. When a number \( a \equiv b \pmod{m} \), it means that \( a - b \) is divisible by \( m \). This concept is crucial in determining how combinations of powers of numbers like 3, 5, and 7 can yield numbers congruent to 1 modulo 4.
By checking each prime factor separately, we can see how their powers influence divisibility by 4, a simple and systematic tool in number theory.

Powers of Integers

When dealing with powers of integers, the key is to understand how raising a number to an exponent changes its behavior under a certain modulus. For numbers in our exercise—\( 3, 5, \) and \( 7 \)—each has a unique relationship with modulo 4 when raised to various powers.

• For \( 3^p \), the number is congruent to 3 modulo 4 unless \( p \) is even, in which case it becomes congruent to 1 modulo 4, because \( 3^2 \equiv 1 \pmod{4} \).
• \( 5^q \) is always congruent to 1 modulo 4, regardless of \( q \) since 5 itself is already congruent to 1 modulo 4.
• \( 7^r \) mimics the behavior of 3, being congruent to 3 modulo 4 unless \( r \) is even, as \( 7^2 \equiv 1 \pmod{4} \).

Understanding these power modulations allows us to predict the form of divisors and solve congruence problems.

Modular Arithmetic

Modular arithmetic, often termed arithmetic of remainders, is a cornerstone of number theory. In this problem, it helps us determine the behavior of powers of integers with respect to a modulus—in this case, 4.
The core principle is working with remainders. For example, since 5 modulo 4 is 1, and multiplying 1 by itself any number of times remains 1, it's clear that \( 5^q \equiv 1 \pmod{4} \) for any integer \( q \).
By applying modular arithmetic to each factor and their respective powers, we craft rules, such as requiring even powers for 3 and 7 to get a remainder of 1 when divided by 4. This technique simplifies complex divisibility and congruence problems into manageable parts.

Divisors and Properties

The study of divisors is crucial in number theory. It involves understanding not just what divides a given number, but how those divisors relate to one another and to given forms, like \( 4\lambda + 1 \).
In this exercise, each integer divisor of \( n \) follows a pattern due to the constraints from the powers of the prime bases.

• For divisors to fit the form \( 4\lambda + 1 \), the total product of the divisors must yield a remainder of 1 when divided by 4.
• This form highlights the inherent property that for divisors to match this pattern, certain combinations of even and odd powers are necessitated, such as even combinations for 3 and 7, and free choice for 5.

Understanding these properties aids in predicting potential divisibility configurations and confirms the validity of developed congruence models.