Missing Digit

Number Theory Level 2

$[ N (N+1) (N+2) ] ^2 = \overline{29{ \color{#D61F06} {A} }4656}$

The above equation is true for some integer $N$ and digit $A$ .

What is the missing digit ${ \color{#D61F06} A}$ ?

5 6 2 4

1 solution

Jesse Nieminen
Oct 26, 2016

The LHS is divisible by $9$ , and hence is the RHS which means that the answer is $4$ .

Show that these are true to understand the solution:

Product of any $n$ consecutive integers is divisible by $n!$ .
If $a \mid x \Rightarrow a^2 \mid x^2$ .
An integer is divisible by $9$ if and only if the sum of its digits in base $10$ is divisible by $9$ .

Ah yes!

N = 11 in the problem. I wanted to keep it small, but not too guessable.

Chung Kevin - 4 years, 7 months ago