Find the smallest values for a and b so that 21 base a equals 25 base b.

21 using base a

= 2a + 1
25 using base b
= 2b+5

2a+1 = 2b+5
2a = 2b+4
a = b+2

in any base system for numbers, the base > 1
so the smallest value of b = 2
then a = 4

The smallest values are a=4, b=2

check:
21 base 4 = 2(4)+1 = 9
25 base 2 = 2(2) + 5 = 9

To find the smallest values for a and b such that 21 in base a equals 25 in base b, we can set up the equation:

21 base a = 25 base b

First, let's convert the numbers to their decimal values:

21 base a = 2*a^1 + 1*a^0 = 2a + 1

25 base b = 2*b^1 + 5*b^0 = 2b + 5

Now, we can set these decimal expressions equal to each other:

2a + 1 = 2b + 5

To find the smallest values for a and b, we can start by assuming a and b to be positive integers and solve the equation.

Subtracting 2b from both sides:

2a - 2b + 1 = 5

Rearranging the terms:

2a - 2b = 4

Dividing both sides by 2:

a - b = 2

Since a and b are positive integers, we can try different values for a and b until we find the smallest solution that satisfies the equation.

Let's start with the smallest possible values for a and b: a = 3 and b = 1.

Substituting these values into the equation:

a - b = 2

3 - 1 = 2

Since the equation holds true, a = 3 and b = 1 are valid values that satisfy the equation. However, we need to find the smallest values for a and b.

Let's try a = 4 and b = 2:

a - b = 2

4 - 2 = 2

Again, the equation holds true. So, a = 4 and b = 2 are valid values that satisfy the equation.

However, we need to find the smallest values for a and b. Since a must be the smallest possible value, we can conclude that a = 3 and b = 1 are indeed the smallest values that satisfy the equation 21 base a = 25 base b.