2021 AMC Fall 10A Problem 8

P1-10~3 minPrint

Bases & Digits Diophantine Equations Organized listing / direct enumeration Set up the equation/formula and compute; no special trick needed

A two-digit positive integer is said to be $\emph{cuddly}$ if it is equal to the sum of its nonzero tens digit and the square of its units digit. How many two-digit positive integers are cuddly?

Show solution without answering

Key insight

Write the number as 10a + b. Cuddly means 9a = b(b - 1). Checking all ten units digits, only b = 9, a = 8 works.

Solution

1. Write the number with its digits. Call the tens digit $a$ (from $1$ to $9$ ) and the units digit $b$ (from $0$ to $9$ ). For example, $47$ is $10 \times 4 + 7$ . So the number is $10a + b$ .
2. Write the rule as an equation. Cuddly means the number equals $a + b^2$ :
$10a + b = a + b^2.$
3. Simplify. Subtract $a$ and $b$ from both sides:
$9a = b^2 - b = b(b - 1).$
So $b(b-1)$ must equal $9$ times the tens digit: one of $9, 18, 27, \ldots, 81$ .
4. Check every units digit. There are only ten choices for $b$ :

$b$	0	1	2	3	4	5	6	7	8	9
$b(b-1)$	0	0	2	6	12	20	30	42	56	72

Only $72$ is on the list $9, 18, \ldots, 81$ . It comes from $b = 9$ , and $9a = 72$ gives $a = 8$ . (The $0$ s would need $a = 0$ , which is not a tens digit.)
Check. $8 + 9^2 = 8 + 81 = 89$ . It works.

There is exactly one cuddly number, $89$ .

The answer is $\boxed{\textbf{(B)}\ 1}$ .

Why this works

Writing a two-digit number as $10a + b$ turns a word rule about digits into an equation. Here the equation leaves only ten cases, so checking them all is fastest and leaves no doubt. You could also notice that $b$ and $b - 1$ are next to each other, so they share no factor of $3$ , which means one of them must be a multiple of $9$ by itself. That again points straight to $b = 9$ .

Concise solution

A shorter write-up, the way contest solutions are usually written. It may take a different route from the solution above.

Let the number have tens digit $a$ ( $1 \le a \le 9$ ) and units digit $b$ ( $0 \le b \le 9$ ), so its value is $10a + b$ . Cuddly means
$10a + b = a + b^2 \quad\Longrightarrow\quad 9a = b^2 - b = b(b-1).$

So $b(b-1)$ must be a positive multiple of $9$ with $b \le 9$ . The values of $b(b-1)$ for $b = 0, 1, \dots, 9$ are $0, 0, 2, 6, 12, 20, 30, 42, 56, 72$ . The only multiples of $9$ are $0$ (from $b = 0$ or $1$ ) and $72$ (from $b = 9$ ). But $9a = 0$ forces $a = 0$ , which is not allowed for a tens digit. The remaining case $b = 9$ gives $9a = 72$ , so $a = 8$ .

The unique cuddly number is $89$ ; indeed $8 + 9^2 = 89$ .

The answer is $\boxed{\textbf{(B)}\ 1}$ .

Review in the AoPS books

Introduction to Number Theory, Chapter 8: Base Numbers
Introduction to Number Theory, Chapter 7: Algebra With Integers
Prealgebra, Chapter 15: Problem-Solving Strategies (§15.2)
Introduction to Number Theory, Chapter 9: Base Number Arithmetic