" Show that the integers m = 3* . 568 and n = 3k. 638, where k > 0, satisfy simultaneously t(m) = t(n), σ(m) = σ (n), and %3D Þ(m) = p(n) %3D
" Show that the integers m = 3* . 568 and n = 3k. 638, where k > 0, satisfy simultaneously t(m) = t(n), σ(m) = σ (n), and %3D Þ(m) = p(n) %3D
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![**Problems 7.2**
1. Calculate \( \phi(1001) \), \( \phi(5040) \), and \( \phi(36,000) \).
2. Verify that the equality \( \phi(n) = (\phi(n+1) = \phi(n+2)) \) holds when \( n = 5186 \).
3. Show that the integers \( n = 3^k \cdot 568 \) and \( n = 3^k \cdot 638 \), where \( k \geq 0 \), satisfy simultaneously:
\[
\tau(m) = \tau(n), \quad \sigma(m) = \sigma(n), \quad \text{and} \quad \phi(m) = \phi(n)
\]
4. Establish each of the assertions below:
(a) If \( n \) is an odd integer, then \( \phi(2n) = \phi(n) \).
(b) If \( n \) is an even integer, then \( \phi(2n) = 2\phi(n) \).
(c) \( \phi(3n) = 3\phi(n) \) if and only if 3 | \( n \).
(d) \( \phi(3n) = 2\phi(n) \) if and only if 3 \(\nmid n\).
(e) \( \phi(n) = n/2 \) if and only if \( n = 2^k \) for some \( k \geq 1 \).
*Hint:* Write \( n = 2^k N \), where \( N \) is odd, and use the condition \( \phi(n) = n/2 \) to show that \( N = 1 \).
5. Prove that the equation \( \phi(n) = \phi(n + 2) \) is satisfied by \( n = 2(2p - 1) \) whenever \( p \) and \( 2p - 1 \) are both odd primes.
6. Show that there are infinitely many integers \( n \) for which \( \phi(n) \) is a perfect square.
*Hint:* Consider the integers \( n = 2^{2k+1} \](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fadf70025-15e9-4d93-968e-9d86049b61bd%2F3e51766b-6365-4504-89ff-577f67fd27dc%2Fjwv6jgn.jpeg&w=3840&q=75)
Transcribed Image Text:**Problems 7.2**
1. Calculate \( \phi(1001) \), \( \phi(5040) \), and \( \phi(36,000) \).
2. Verify that the equality \( \phi(n) = (\phi(n+1) = \phi(n+2)) \) holds when \( n = 5186 \).
3. Show that the integers \( n = 3^k \cdot 568 \) and \( n = 3^k \cdot 638 \), where \( k \geq 0 \), satisfy simultaneously:
\[
\tau(m) = \tau(n), \quad \sigma(m) = \sigma(n), \quad \text{and} \quad \phi(m) = \phi(n)
\]
4. Establish each of the assertions below:
(a) If \( n \) is an odd integer, then \( \phi(2n) = \phi(n) \).
(b) If \( n \) is an even integer, then \( \phi(2n) = 2\phi(n) \).
(c) \( \phi(3n) = 3\phi(n) \) if and only if 3 | \( n \).
(d) \( \phi(3n) = 2\phi(n) \) if and only if 3 \(\nmid n\).
(e) \( \phi(n) = n/2 \) if and only if \( n = 2^k \) for some \( k \geq 1 \).
*Hint:* Write \( n = 2^k N \), where \( N \) is odd, and use the condition \( \phi(n) = n/2 \) to show that \( N = 1 \).
5. Prove that the equation \( \phi(n) = \phi(n + 2) \) is satisfied by \( n = 2(2p - 1) \) whenever \( p \) and \( 2p - 1 \) are both odd primes.
6. Show that there are infinitely many integers \( n \) for which \( \phi(n) \) is a perfect square.
*Hint:* Consider the integers \( n = 2^{2k+1} \
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