1+2^2+3^3+4^4+.....+10^10
when is is divided by 5 what is the remainder!!
show working,please have been trying that i could not,get it
thats the real question
Sum(1) = 1 = 1mod5
sum(2) = 1 + 2^2 = 5 = 0 mod5
sum(3) = 5 + 3^3 = 32 = 2 mod5
sum(4) = 32 + 4^4 = 288 = 3 mod5
To divide by 5 all we need to do is look at the last digit.
If the digit is form 0 to 4, then that digit is the remainder when dividing by 5
If the digit is from 5 to 9, the remainder is (that digit - 5)
e.g. 245678 leaves a remainder of 3
on my calculator continuing this until
Sum(8) = 873612 + 8^8 = 17650828
on the next step, many calculators will kick into scientific notation and lose the unit digits.
however, 9^9 ends in ...9 , so adding this to
Sum(9) ends in ..... 7
and we know that 10^10 ends in 0
so
Sum(0) ends in ....0
so the remainder when divided by 5 is 0
btw, the notation
288 = 3 mod5
means that the remainder is 3 when 288 is divided by 5
To find the remainder when the expression 1 + 2^2 + 3^3 + 4^4 + ... + 10^10 is divided by 5, we need to evaluate the expression and then calculate the remainder.
First, let's expand the expression:
1^1 + 2^2 + 3^3 + 4^4 + 5^5 + 6^6 + 7^7 + 8^8 + 9^9 + 10^10
Now, we can calculate each term and simplify them:
= 1 + 4 + 27 + 256 + 3125 + 46656 + 823543 + 16777216 + 387420489 + 10000000000
Next, we sum up all these terms:
= 10000213455
Finally, we divide this result by 5 and find the remainder:
Remainder = 10000213455 % 5
To calculate this, we divide 10000213455 by 5 and find the remainder:
Remainder = 0
Therefore, when the expression 1 + 2^2 + 3^3 + 4^4 + ... + 10^10 is divided by 5, the remainder is 0.