Write the base 10 number 945 s a base 6 number and as a base 12 number. then, using these numbers as base 10 numbers, find the positive difference between them.
ABC=495. 495+459=954
To convert a base 10 number to a different base, you need to divide the number continuously by the desired base until the quotient is 0, keeping track of the remainders at each step. Then, the remainders in reverse order represent the base representation of the number.
Let's start with converting 945 to base 6:
Step 1: Divide 945 by 6
Quotient: 157 Remainder: 3
Step 2: Divide 157 by 6
Quotient: 26 Remainder: 1
Step 3: Divide 26 by 6
Quotient: 4 Remainder: 2
Step 4: Divide 4 by 6
Quotient: 0 Remainder: 4
The remainders in reverse order are 4321. Therefore, 945 in base 10 is equal to 4321 in base 6.
Now, let's convert 945 to base 12:
Step 1: Divide 945 by 12
Quotient: 78 Remainder: 9
Step 2: Divide 78 by 12
Quotient: 6 Remainder: 6
Step 3: Divide 6 by 12
Quotient: 0 Remainder: 6
The remainders in reverse order are 966. Therefore, 945 in base 10 is equal to 966 in base 12.
To find the positive difference between these two numbers, we need to convert them both back to base 10 and then subtract:
Converting 4321 to base 10:
4321 = (4 * 6^3) + (3 * 6^2) + (2 * 6^1) + (1 * 6^0)
= 864 + 108 + 12 + 1
= 985
Converting 966 to base 10:
966 = (9 * 12^2) + (6 * 12^1) + (6 * 12^0)
= 1296 + 72 + 6
= 1374
Now, finding the positive difference:
1374 - 985 = 389
Therefore, the positive difference between 4321 (in base 6) and 966 (in base 12), when treated as base 10 numbers, is 389.