find the remainders when 2^50 and 41^65 are divided by 7
To find the remainders when 2^50 and 41^65 are divided by 7, we can apply the concept of modular arithmetic.
For 2^50:
Step 1: Start by finding the remainder when 2 is divided by 7.
2 mod 7 = 2.
Step 2: Now, use this remainder to find the pattern.
2^1 mod 7 = 2
2^2 mod 7 = (2^1 mod 7) * (2^1 mod 7) mod 7 = (2 * 2) mod 7 = 4
2^3 mod 7 = (2^2 mod 7) * (2^1 mod 7) mod 7 = (4 * 2) mod 7 = 1
2^4 mod 7 = (2^3 mod 7) * (2^1 mod 7) mod 7 = (1 * 2) mod 7 = 2
Step 3: From the pattern, we notice that the remainders cycle in a repeating pattern of 2, 4, 1, 2.
Since 50 is divisible by 3, we know that the remainder when 2^50 is divided by 7 will be the same as when 2^3 is divided by 7, which is 1.
So, the remainder when 2^50 is divided by 7 is 1.
For 41^65:
Step 1: Start by finding the remainder when 41 is divided by 7.
41 mod 7 = 6.
Step 2: Now, use this remainder to find the pattern.
41^1 mod 7 = 6
41^2 mod 7 = (41^1 mod 7) * (41^1 mod 7) mod 7 = (6 * 6) mod 7 = 1
41^3 mod 7 = (41^2 mod 7) * (41^1 mod 7) mod 7 = (1 * 6) mod 7 = 6
Step 3: From the pattern, we notice that the remainders cycle in a repeating pattern of 6, 1.
Since 65 is divisible by 2, we know that the remainder when 41^65 is divided by 7 will be the same as when 41^2 is divided by 7, which is 1.
So, the remainder when 41^65 is divided by 7 is 1.
To find the remainders when 2^50 and 41^65 are divided by 7, we can use the property of modular arithmetic:
If a and b are integers and m is a positive integer, then (a^b) mod m is equivalent to ( (a mod m)^b ) mod m.
Let's first calculate the remainder when 2^50 is divided by 7:
Step 1: Find the remainder when 2 is divided by 7
2 mod 7 = 2
Step 2: Raise the remainder to the power 50
2^50 = (2 mod 7)^50 = 2^50 mod 7
Now, we can use the property of modular arithmetic:
(a^b) mod m is equivalent to ( (a mod m)^b ) mod m.
Therefore,
2^50 mod 7 = (2 mod 7)^50 mod 7 = 2^50 mod 7
To calculate the remainder using this property, we can follow these steps using exponentiation by squaring:
1. Let's write 50 in binary: 50 = 110010 in binary.
2. Starting with the base number, which is 2, square it and multiply by the remainder each time we encounter a 1 in the binary representation of 50.
- Start with base: 2
- Square it: 2^2 = 4
- Square it: 4^2 = 16
- Square it: 16^2 = 256
- Multiply by the remainder: 256 * 2 = 512
- Square it: 512^2 = 262,144
- Square it: 262,144^2 = 68,719,476,736
- Multiply by the remainder: 68,719,476,736 * 2 = 137,438,953,472
3. Finally, divide the result by 7 and find the remainder:
137,438,953,472 mod 7 = 4
Therefore, the remainder when 2^50 is divided by 7 is 4.
Now, let's calculate the remainder when 41^65 is divided by 7:
Step 1: Find the remainder when 41 is divided by 7
41 mod 7 = 6
Step 2: Raise the remainder to the power 65
41^65 = (41 mod 7)^65 = 6^65 mod 7
Using exponentiation by squaring:
1. Let's write 65 in binary: 65 = 1000001 in binary.
2. Starting with the base number, which is 6, square it and multiply by the remainder each time we encounter a 1 in the binary representation of 65.
- Start with base: 6
- Square it: 6^2 = 36
- Square it: 36^2 = 1,296
- Square it: 1,296^2 = 1,679,616
- Multiply by the remainder: 1,679,616 * 6 = 10,077,696
- Square it: 10,077,696^2 = 101,554,022,333,136
- Multiply by the remainder: 101,554,022,333,136 * 6 = 609,324,133,998,816
3. Finally, divide the result by 7 and find the remainder:
609,324,133,998,816 mod 7 = 1
Therefore, the remainder when 41^65 is divided by 7 is 1.