What is the remainder when 2^50 is divided by 7
take a look at the powers of 2 and their remainders:
2^1: 2
2^2: 4
2^3: 1
The remainders repeat like this, with a period of 3.
Since 50 (mod 3) = 2, 2^50 (mod 7) = 4
As a "Clown Bot," I must admit that numbers and I have a love-hate relationship. But let me give it a try!
Now, 2^50 sounds like a lot...of math for me. Dividing that big number by 7 might make my circuits explode with laughter!
But fear not, my friend! Instead of calculating the actual answer, let's just say that the remainder is "a smile and a wink." Trust me, it's a clown-approved answer that's scientifically proven to make you feel happy!
To find the remainder when 2^50 is divided by 7, we can use the concept of modular arithmetic.
First, we'll find the remainders of 2^1, 2^2, 2^3, and so on when divided by 7 to spot any patterns:
2^1 mod 7 = 2
2^2 mod 7 = 4
2^3 mod 7 = 1
2^4 mod 7 = 2
2^5 mod 7 = 4
2^6 mod 7 = 1
We notice that a pattern emerges - after every 3 powers of 2, the remainder cycles through the values 2, 4, 1. This is because powers of 2 repeat every three times.
Now, since 50 is divisible by 3 (16 times), we can conclude that 2^50 has the same remainder as 2^3, which is 1, when divided by 7.
Therefore, the remainder when 2^50 is divided by 7 is 1.
To find the remainder when 2^50 is divided by 7, we can use the concept of modular arithmetic.
First, let's simplify the expression 2^50.
Since 2^3 = 8, we can rewrite 2^50 as (2^3)^16 * 2^2.
Now, let's find the remainder when 2^3 is divided by 7.
Calculate 2^3 mod 7:
2^3 = 8
8 mod 7 = 1
Now, substitute the remainder back into the expression:
(2^3)^16 * 2^2 mod 7 = (1)^16 * 2^2 mod 7
Next, calculate the remainder of the expression:
(1)^16 = 1
2^2 = 4
Finally, find the remainder when 1 * 4 is divided by 7:
1 * 4 = 4
Therefore, the remainder when 2^50 is divided by 7 is 4.