Homework 5: Number Theory

Problem 1
 

Find all positive integers \( n \) such that \( n(n+1) \) is a perfect square.

Problem 2
 

Assume that \( x \) and \( y \) are non-negative integers such that \( 15x+11y \) is divisible by \( 37 \). Prove that \( 7x+15y \) is divisible by \( 37 \).

Problem 3
 

There are \( n \) books in a library. If books are to be arranged in boxes with \( 7 \) books in each box, then \( 5 \) books remain. If they are arranged with \( 9 \) books in each box, then \( 3 \) books remain, and if they are arranged with \( 11 \) books in each box, then \( 7 \) books remain. What is the smallest possible value for \( n \).

Problem 4
 

Assume that \( n_1 \), \( n_2 \), \( \dots \), \( n_k \) are positive integers whose greatest common divisor is equal to \( d \). Prove that there exist integers \( \alpha_1 \), \( \alpha_2 \), \( \dots \), \( \alpha_k \) such that \[ \alpha_1n_1+\alpha_2n_2+\cdots+\alpha_kn_k=d.\]

Problem 5
 

Let \( n\geq 3 \) be an odd integer. Prove that every integer \( l \) satisfying \( 1\leq l\leq n \) can be represented as a sum or difference of two integers each of which is less than \( n \) and relatively prime to \( n \).

Problem 6
 

Prove that there is no positive integer \( n \) for which \( n^5 \) can be written as a product of six consecutive positive integers.

Problem 7
 

Let \( n\geq 5 \) be a natural number. Prove that the following two statements are equivalent:

  • (a) Neither of the numbers \( n \) and \( n+1 \) is prime.

  • (b) The closest integer to \( \displaystyle\frac{(n-1)!}{n^2+n} \) is even.

Remark. If \( k\in\mathbb Z \), the closest integer to \( k+\frac12 \) is \( k+1 \).

Problem 8
 

Does there exist \( k,n\in\mathbb N \) such that \( k\geq 2 \) and for which the set \( \{n,n+1,n+2, \dots, n+101\} \) can be partitioned into \( k \) disjoint subsets all of which have equal products of elements?


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