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3. Prove that if n is a triangular number, then so are 9n + 1, 25n + 3, and 49n+ 6. Hint: Recall that in the formula for finding the n-th triangular number, the numerator is a product of two consecutive integers.

3. Prove that if n is a triangular number, then so are 9n + 1, 25n + 3, and 49n+ 6. Hint: Recall that in the formula for finding the n-th triangular number, the numerator is a product of two consecutive integers.

Algebra and Trigonometry (6th Edition)
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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Pls. answer number 3 only and provide complete solution.

1. A primitive Pythagorean triple is an ordered tripe of integers (x, y, z) such that x² + y? = z?, where x, y and z are pairwisely relatively prime integers. Determine exactly 10 primitive Pythagorean triples and be able to exhibit that they satisfy the given equation. 2. Solve the system of congruence 3x + 7y = 10 ( mod 16) 5x + 2y = 9( mod 16) . Hint: Eliminate x by multiplying each congruence a suitable constant and then adding them to form a linear congruence containing only y as a variable. Likewise, eliminate y by multiplying each congruence a suitable constant and then adding them to form a linear congruence containing only x as a variable. 3. Prove that if n is a triangular number, then so are 9n + 1, 25n + 3, and 49n + 6. Hint: Recall that in the formula for finding the n-th triangular number, the numerator is a product of two consecutive integers. 4. A palindrome number is a number that remains the same when its digits are reversed. For example, the following numbers are palindromes: 7 22 131 5665 10 901 480 084. Show that a palindrome with an even number of digits is divisible by 11. Hint: Read your notes.
Transcribed Image Text:1. A primitive Pythagorean triple is an ordered tripe of integers (x, y, z) such that x² + y? = z?, where x, y and z are pairwisely relatively prime integers. Determine exactly 10 primitive Pythagorean triples and be able to exhibit that they satisfy the given equation. 2. Solve the system of congruence 3x + 7y = 10 ( mod 16) 5x + 2y = 9( mod 16) . Hint: Eliminate x by multiplying each congruence a suitable constant and then adding them to form a linear congruence containing only y as a variable. Likewise, eliminate y by multiplying each congruence a suitable constant and then adding them to form a linear congruence containing only x as a variable. 3. Prove that if n is a triangular number, then so are 9n + 1, 25n + 3, and 49n + 6. Hint: Recall that in the formula for finding the n-th triangular number, the numerator is a product of two consecutive integers. 4. A palindrome number is a number that remains the same when its digits are reversed. For example, the following numbers are palindromes: 7 22 131 5665 10 901 480 084. Show that a palindrome with an even number of digits is divisible by 11. Hint: Read your notes.
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