Find the value of 321 base 5 × 21 base 5
To multiply these two numbers together, we can use the same method as multiplying in base 10. Let's break it down step by step:
```
321
x 21
-------
```
First, we multiply the units digit of the second number (1) with every digit in the first number:
```
321
x 21
-------
321
+ 3210
-------
```
Now we multiply the tens digit of the second number (2) with every digit in the first number:
```
321
x 21
-------
321
+ 3210
+642
-------
```
Finally, we multiply the hundreds digit of the second number (0) with every digit in the first number:
```
321
x 21
-------
321
+ 3210
+ 642
-------
321
+ 3210
+ 6420
-------
```
By summing all the partial products, we get:
```
321
x 21
-------
10011
+ 32100
+ 32100
-------
100211
-------
```
Therefore, the value of 321 base 5 × 21 base 5 is 100211 base 5.
To find the product of two numbers in base 5, we can multiply them just like we would in base 10. Let's calculate step-by-step:
Step 1: Write down the numbers 321 and 21 vertically, aligning them by their place values.
321
× 21
Step 2: Start by multiplying the units digits (1 × 1).
321
× 21
------
...
+...
The product of 1 multiplied by 1 is 1, so we write down 1 as the units digit.
321
× 21
------
1
+...
Step 3: Next, multiply the tens digit of the bottom number (2) by both digits of the top number (1 and 2). Write down the results below each other.
321
× 21
------
1
+4
------
The product of 2 multiplied by 1 is 2, and the product of 2 multiplied by 2 is 4.
321
× 21
------
1
+4
------
2
Step 4: Finally, multiply the hundreds digit of the bottom number (3) by both digits of the top number (1 and 2). Write down the results below each other.
321
× 21
------
1
+4
------
2
+6
------
The product of 3 multiplied by 1 is 3, and the product of 3 multiplied by 2 is 6.
321
× 21
------
1
+4
------
2
+6
------
3
Step 5: Add up the partial products.
1
+4
------
2
+6
------
13
The final product is 13 base 5.
Therefore, 321 base 5 × 21 base 5 = 13 base 5.
To find the value of 321 base 5 × 21 base 5, we first need to convert the numbers to base 10, perform the multiplication, and then convert the result back to base 5.
Converting 321 base 5 to base 10:
To convert a number from base 5 to base 10, we multiply each digit by 5 raised to the power of its position and sum the results.
321 base 5 = (3 × 5^2) + (2 × 5^1) + (1 × 5^0)
= (3 × 25) + (2 × 5) + (1 × 1)
= 75 + 10 + 1
= 86
Converting 21 base 5 to base 10:
21 base 5 = (2 × 5^1) + (1 × 5^0)
= (2 × 5) + (1 × 1)
= 10 + 1
= 11
Now, we can perform the multiplication in base 10:
(86) × (11) = 946
Converting the result back to base 5:
To convert a base 10 number to base 5, we repeatedly divide the number by 5 and keep track of the remainders from right to left. The final result is the remainders read from left to right.
946 ÷ 5 = 189 remainder 1
189 ÷ 5 = 37 remainder 4
37 ÷ 5 = 7 remainder 2
7 ÷ 5 = 1 remainder 2
1 ÷ 5 = 0 remainder 1
Reading the remainders from left to right, the result is 12412 base 5.
Therefore, the value of 321 base 5 × 21 base 5 is 12412 base 5.