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5 Hc. Is n' divisible by 20? Explain. H d. What is the least positive integer m so that 8,424 m is a perfect square?

5 Hc. Is n' divisible by 20? Explain. H d. What is the least positive integer m so that 8,424 m is a perfect square?

Advanced Engineering Mathematics
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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**Transcription for Educational Content:** **Divisibility Rules:** For an integer to be divisible by 3, the sum of its digits must be divisible by 3. For divisibility by 9, the sum of its digits must be divisible by 9. An integer is divisible by 5 if its right-most digit is 5 or 0. For divisibility by 4, the number formed by its right-most two digits must be divisible by 4. *Check the following integers for divisibility by 3, 4, 5, and 9:* a. 637,425,403,705,125 b. 12,858,306,120,312 c. 517,924,440,926,512 d. 14,328,083,360,232 **Unique Factorization Theorem:** 37. Use the unique factorization theorem to write the following integers in standard factored form. a. 1,176 b. 5,733 c. 3,675 38. Let \( n = 8,424 \). a. Write the prime factorization for \( n \). b. Write the prime factorization for \( n^5 \). c. Is \( n^5 \) divisible by 20? Explain. d. What is the least positive integer \( m \) so that 8,424\( \times m \) is a perfect square? 39. Suppose that in standard factored form \( a = p_1^{e_1} p_2^{e_2} \cdots p_k^{e_k} \), where \( k \) is a positive integer; \( p_1, p_2, \ldots, p_k \) are prime numbers; and \( e_1, e_2, \ldots, e_k \) are positive integers. a. What is the standard factored form for \( a^3 \)? b. Find the least positive integer \( k \) such that \( 2^4 \cdot 3^5 \cdot 7 \cdot 11^2 \cdot k \) is a perfect cube (that is, it equals an integer to the third power). Write the resulting product as a perfect cube.
Transcribed Image Text:**Transcription for Educational Content:** **Divisibility Rules:** For an integer to be divisible by 3, the sum of its digits must be divisible by 3. For divisibility by 9, the sum of its digits must be divisible by 9. An integer is divisible by 5 if its right-most digit is 5 or 0. For divisibility by 4, the number formed by its right-most two digits must be divisible by 4. *Check the following integers for divisibility by 3, 4, 5, and 9:* a. 637,425,403,705,125 b. 12,858,306,120,312 c. 517,924,440,926,512 d. 14,328,083,360,232 **Unique Factorization Theorem:** 37. Use the unique factorization theorem to write the following integers in standard factored form. a. 1,176 b. 5,733 c. 3,675 38. Let \( n = 8,424 \). a. Write the prime factorization for \( n \). b. Write the prime factorization for \( n^5 \). c. Is \( n^5 \) divisible by 20? Explain. d. What is the least positive integer \( m \) so that 8,424\( \times m \) is a perfect square? 39. Suppose that in standard factored form \( a = p_1^{e_1} p_2^{e_2} \cdots p_k^{e_k} \), where \( k \) is a positive integer; \( p_1, p_2, \ldots, p_k \) are prime numbers; and \( e_1, e_2, \ldots, e_k \) are positive integers. a. What is the standard factored form for \( a^3 \)? b. Find the least positive integer \( k \) such that \( 2^4 \cdot 3^5 \cdot 7 \cdot 11^2 \cdot k \) is a perfect cube (that is, it equals an integer to the third power). Write the resulting product as a perfect cube.
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