Question

Let $L_n$ denote the least common multiple of the numbers $1,$ $2,$ $3,$ $\ldots,$ $n,$ and let $h$ be the unique positive integer such that $\dfrac{1}{1} + \dfrac{1}{2} + \dfrac{1}{3} + \cdots + \dfrac{1}{17} = \dfrac{h}{L_{17}}$ What is the remainder when $h$ is divided by $17?$

$1$

$3$

$5$

$7$

$9$

Answer: C

Concepts:modular arithmetic least common multiple prime factorization

Difficulty rating: 2150

Small Hint:

Work modulo $17$ after multiplying by the least common multiple

Big Hint:

All terms except the one from $\frac{1}{17}$ vanish modulo $17$

Solution:

Multiplying the harmonic sum by $L_{17},$ we get $h=\sum_{i=1}^{17}\frac{L_{17}}{i}.$

For $1\le i\le16,$ the term $\frac{L_{17}}{i}$ is still divisible by $17,$ so these terms contribute $0\pmod{17}.$

Thus $h\equiv \frac{L_{17}}{17}\pmod{17}.$ The least common multiple $L_{17}$ contains the prime-power factors $16,9,5,7,11,13,17,$ so $\begin{aligned} \frac{L_{17}}{17} &\equiv 16\cdot9\cdot5\cdot7\cdot11\cdot13 \\ &\pmod{17}. \end{aligned}$

Reducing modulo $17,$ this is $(-1)\cdot9\cdot5\cdot7\cdot11\cdot13$ $\equiv5\pmod{17}.$

Thus, C is the correct answer.

2022 AMC 10A Problem 19

Answer: C

Difficulty rating: 2150

Small Hint:

Big Hint:

Solution:

Problem 19 in Other Years