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.