[Soln] 2014 Singapore MO (Senior Rd 1) Problem 17

2014 SMO (Senior-Rd1) 17. Let n be a positive integer such that 12n^2 + 12n + 11 is a 4-digit number with all 4 digits equal. Determine the value of n.

Let 12n^2 + 12n + 11 = \overline{aaaa} for some digit a. A brute-force method would be to let a run through 1 to 9: for each value of a, solve the quadratic equation in n and check if the resulting value of n is a positive integer. The problem with this method is that there will be big numbers encountered in the process, which would slow down the problem solving process.

Let’s try to make use of the given expression. What stands out is the 2 coefficients 12: how nice it would be if the last coefficient was 12 as well! Instead, it is 11. What this means is that \boldsymbol{\overline{aaaa}} must be 1 less than a multiple of 12. We can use moduli to narrow down the possibilities for \boldsymbol{a}.

For instance, we know that \overline{aaaa} must be odd, hence

a = 1, 3, 5, 7 \text{ or } 9.

Why stop at considering just \overline{aaaa} \mod 2? Taking \mod 3, since

12n^2 + 12n + 11 \equiv 2 \hspace{1em} (\text{mod } 3),

we must have

\begin{aligned} \overline{aaaa} &\equiv 2 &\mod 3, \\  4a &\equiv 2 &\mod 3. \end{aligned}

(Exercise: Prove that n \equiv \text{ digits of } n \mod 3 for all positive integers n.) The only odd value of a that satisfies the congruence above is a = 5. Hence, we must have

\begin{aligned} 12n^2 + 12n + 11 &= 5555, \\  12n^2 + 12n &= 5544, \\  n^2 + n &= 462, \\  (n+22)(n-21) &= 0. \end{aligned}

Since n is a positive integer, we must have n = 21. The answer is 21.

Advertisements
This entry was posted in Grade 10, Singapore and tagged , , . Bookmark the permalink.

Leave a Reply

Please log in using one of these methods to post your comment:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s