Find a base 7 three-digit number which has its digits reversed when expressed in base 9.

(You do not need to indicate the base with a subscript for this answer.)

well, using base 10 for now, if the number has base7 digits abc, we need

49a+7b+c = 81c+9b+a

c = 3a/5 - b/40
40c = 24a-b
If c=1, a=2,b=8 Nope
If c=2, a=4,b=16 Nope
If c=3, a=5,b=0

5037 = 3059

check:
49*5+3 = 81*3+5
yes

That would be wrong, sorry.

thats actually correct you know?

To find a base 7 three-digit number that has its digits reversed when expressed in base 9, we can follow a step-by-step process:

1. Convert the base 7 number to base 10: Let's assume the three-digit number in base 7 is represented as $xyz_7$.

The rightmost digit 'z' represents the units place, 'y' represents the tens place, and 'x' represents the hundreds place.

To convert the number to base 10, we need to multiply each digit by the corresponding power of 7 and then sum them up:

$xyz_7 = x \cdot 7^2 + y \cdot 7^1 + z \cdot 7^0$

2. Reverse the order of the digits: Now, we need to write $xyz_7$ in reverse order. So, the reversed three-digit number in base 10 is $zyx_{10}$.

3. Convert the reversed number to base 9: Next, we convert the reversed number in base 10 to base 9. Let's assume the three-digit number in base 9 is represented as $zyx_9$.

To convert the number to base 9, we need to divide it successively by 9 and record the remainders:

$zyx_9 = q_1 \cdot 9^2 + q_2 \cdot 9^1 + q_3 \cdot 9^0$

Where $q_1$ is the quotient of the division of $zyx_{10}$ by 9^2, $q_2$ is the quotient of the division of the remainder of the previous step by 9^1, and $q_3$ is the quotient of the division of the remainder of the previous step by 9^0.

4. Find a base 7 number with reversed digits: Now, search for a number $xyz_7$ that satisfies the condition $zyx_9 = q_1 \cdot 9^2 + q_2 \cdot 9^1 + q_3 \cdot 9^0$.

By trying different combinations of $x$, $y$, and $z$, you will find that there is one unique base 7 three-digit number that satisfies the condition.

And that's how you can find a base 7 three-digit number which has its digits reversed when expressed in base 9.