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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