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## 18S.Fractions Mod p and Wolstenholme’s Theorem

A23 Prove that if

$1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{p-1}$

is expressed as a fraction, where $p>3$ is a prime, then $p^2$ divides the numerator.

A24 Let $p>3$ be a prime number and $k=\lfloor\frac{2p}{3}\rfloor$. Prove that

${p \choose 1}+{p \choose 2}+\cdots+{p \choose k}$

is divisible by $p^2$.

Here is the official solution file: pen18.pdf

## 18. Fractions Mod p and Wolstenholme’s Theorem

A23 Prove that if

$1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{p-1}$

is expressed as a fraction, where $p>3$ is a prime, then $p^2$ divides the numerator.

A24 Let $p>3$ be a prime number and $k=\lfloor\frac{2p}{3}\rfloor$. Prove that

${p \choose 1}+{p \choose 2}+\cdots+{p \choose k}$

is divisible by $p^2$.

## 17S. A Hidden Divisibility

A13 Show that for all prime numbers $p$,

$Q(p) = \prod^{p - 1}_{k = 1}k^{2k - p - 1}$

is an integer.

Here is the official solution file: pen17.pdf.

## 17. A Hidden Divisibility

A13 Show that for all prime numbers $p$,

$Q(p) = \prod^{p - 1}_{k = 1}k^{2k - p - 1}$

is an integer.

## Winter Break

We will be back on February!

C2 The positive integers $a$ and $b$ are such that the numbers $15a+16b$ and $16a-15b$ are both squares of positive integers. What is the least possible value that can be taken on by the smaller of these two squares?
C2 The positive integers $a$ and $b$ are such that the numbers $15a+16b$ and $16a-15b$ are both squares of positive integers. What is the least possible value that can be taken on by the smaller of these two squares?