What is the last two digit of 3243^8897 + 6574^2314 ?
pls give me the explaination as well:)
thank you.
To find the last two digits of the expression 3243^8897 + 6574^2314, we need to break it down into smaller steps.
Step 1: Find the last two digits of 3243^8897.
To find the last two digits of a number raised to a large power, we can look for a pattern among the powers of the number. Let's start by finding the last two digits of the first few powers of 3243:
3243^1 = 3243 (last two digits: 43)
3243^2 = 10,516,249 (last two digits: 49)
3243^3 = 34,073,314,507 (last two digits: 07)
3243^4 = 110,091,642,543 (last two digits: 43)
3243^5 = 356,617,594,349 (last two digits: 49)
By observing the pattern, we notice that the last two digits repeat every 4 powers: 43, 49, 07, 43. Since 8897 is not divisible by 4 (remainder of 1), we know that the last two digits of 3243^8897 will be the same as the last two digits of 3243^1, which is 43.
Step 2: Find the last two digits of 6574^2314.
Similarly, let's find the last two digits of the first few powers of 6574:
6574^1 = 6574 (last two digits: 74)
6574^2 = 43,175,876 (last two digits:76)
6574^3 = 283,972,642,424 (last two digits: 24)
6574^4 = 1,864,164,657,776 (last two digits: 76)
6574^5 = 12,255,610,041,024 (last two digits: 24)
Again, we find that the last two digits repeat every 4 powers: 74, 76, 24, 76. Since 2314 is divisible by 4, we know that the last two digits of 6574^2314 will be the same as the last two digits of 6574^4, which is 76.
Step 3: Add the last two digits of 3243^8897 and 6574^2314.
Now, we can simply add the last two digits calculated in steps 1 and 2:
43 + 76 = 119.
So, the last two digits of 3243^8897 + 6574^2314 is 19.