$a\times b^{a}=xy$

Number Theory Level 3

$a\times b^{a}=xy$

Where xy is a two digit number and $x+y=a\times b$

a is not equal to b

Find xy

The answer is 24.

1 solution

Joshua Lowrance
Feb 21, 2019

We can look at the prime factorization of numbers under 100 (because $xy$ is a two digit number) in the form of $a \times b^a$ , and see if one works.

It is fairly easy to see that $a$ or $b$ cannot equal $1$ , because otherwise $a \times b^a < 10$ , and $xy$ is a two digit number.

$a \times b^a = xy$	$x+y \stackrel{?}{=} a \times b$
$2 \times 3^2 = 18$	$1+8 \neq 2\times 3$
$2 \times 4^2 = 32$	$3+2 \neq 2 \times 4$
$2 \times 5^2 = 50$	$5+0 \neq 2 \times 5$
$2 \times 6^2 = 72$	$7+2 \neq 2 \times 6$
$2 \times 7^2 = 96$	$9+6 \neq 2 \times 7$
$3 \times 2^3 = 24$	$2+4 = 3 \times 2$
$3 \times 3^3 = 81$	$8+1 = 3 \times 3$
$4 \times 2^4 = 64$	$6+4 \neq 4 \times 2$

These are the only two digit numbers under 100 in the form of $a \times b^a$ . The only ones that work are $x=2,y=4,a=3,b=2$ and $x=8,y=1,a=3,b=3$ . However, $a \neq b$ , so therefore $xy =24$ is the only solution.

How is my question?

chakravarthy b - 2 years, 3 months ago

I like your question a lot! I found it very intriguing and I loved the unique solution to it!

Joshua Lowrance - 2 years, 3 months ago

@Joshua Lowrance thanks

chakravarthy b - 2 years, 3 months ago

a × b a = x y a\times b^{a}=xy a × b a = x y

The answer is 24.

1 solution

0 pending reports

$a\times b^{a}=xy$