Determine the sum of the digits of
1) 777 777 777 777^2 - 222 222 222 223^2
Factorize the difference of two squares:
777 777 777 777^2 - 222 222 222 223^2
=(777777777777+222222222223)*(777777777777-222222222223)
=1000000000000*555555555554
=555555555554000000000000
The sum of the digits
= 12*5-1
=59
To determine the sum of the digits of the given expression, let's break it down into smaller steps:
Step 1: Calculate the square of 777 777 777 777.
777 777 777 777^2 = 604 938 271 604 938 271 604 938 271.
Step 2: Calculate the square of 222 222 222 223.
222 222 222 223^2 = 49 382 716 049 382 716 049 382 716 049.
Step 3: Subtract the result of step 2 from the result of step 1.
604 938 271 604 938 271 604 938 271 - 49 382 716 049 382 716 049 382 716 049 = 555 555 555 555 555 555 555 555.
Step 4: Calculate the sum of the digits in the final result.
5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 = 110.
Therefore, the sum of the digits of 777 777 777 777^2 - 222 222 222 223^2 is 110.
To determine the sum of the digits of the given expression, we first need to calculate the actual value of the expression and then find the sum of its digits.
Let's break down the expression into two parts:
1) (777,777,777,777)^2
2) (222,222,222,223)^2
Calculating the first expression:
To find the square of a number, we multiply it by itself. However, multiplying such large numbers manually can be time-consuming. Therefore, we can use a calculator or write a simple Python program to calculate it for us.
In Python, you can calculate the square of a number using the exponentiation operator (**). Thus, we can compute the value of the first expression as follows:
777,777,777,777^2 = (777,777,777,777) ** 2 = 604,938,271,604,938,271,60,493,827,160,493,827,160,493
Now let's calculate the second expression:
Similarly, we find the square of (222,222,222,223):
222,222,222,223^2 = (222,222,222,223) ** 2 = 49,382,716,049,382,716,09,382,716,09,382,716,009
Now, subtracting the two expressions:
604,938,271,604,938,271,60,493,827,160,493,827,160,493 - 49,382,716,049,382,716,09,382,716,09,382,716,009
Performing the subtraction gives us:
604,888,888,888,888,555,551,778,148,148,148,148,484
Finally, let's find the sum of the digits of this number:
To find the sum of the digits, we can add each individual digit together. Again, we can use a calculator or write a simple Python program to calculate it for us.
In Python, you can convert the number to a string and iterate over each character (digit) in the string, converting it back to an integer and adding it to a running total.
Summing the digits of the given number, 604,888,888,888,888,555,551,778,148,148,148,148,484, gives us:
6 + 0 + 4 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 5 + 5 + 5 + 5 + 1 + 7 + 7 + 8 + 1 + 4 + 8 + 1 + 4 + 8 + 1 + 4 + 8 + 1 + 4 + 8 + 4 = 160