Question

The number $695$ is to be written with a factorial base of numeration, that is, $\begin{aligned} 695={}&a_1+a_2\cdot2!+a_3\cdot3!\\ &+\cdots+a_n\cdot n!, \end{aligned}$ where $a_1,$ $a_2,$ $a_3,$ $\ldots,$ $a_n$ are integers such that $0\le a_k\le k,$ and $n!$ means $n(n-1)(n-2)\cdots2\cdot1.$ Find $a_4.$

$0$

$1$

$2$

$3$

$4$

Answer: D

Concepts:number base factorial

Difficulty rating: 1500

Small Hint:

Begin with the largest factorial not exceeding $695$

Big Hint:

After removing the $5!$ contribution, divide the remainder by $4!$

Solution:

Since $5!=120,$ $695=5\cdot120+95.$ Next $4!=24,$ and $95=3\cdot24+23.$ Therefore the coefficient $a_4$ is $3.$

Thus, the correct answer is D.

1961 AMC 12 Problem 35

Answer: D

Difficulty rating: 1500

Small Hint:

Big Hint:

Solution:

Problem 35 in Other Years