The value of x in the set
{0,1,2 ...10}
Satisfying 3-x≡5mod11 is.....???
Note that 5 mod11 = -6 mod 11
so x = 9
3-9+11 = 5
To find the value of x that satisfies the equation 3-x ≡ 5 (mod 11), we can follow these steps:
Step 1: Subtract 3 from both sides of the equation:
3 - x ≡ 5 (mod 11)
- x ≡ 2 (mod 11)
Step 2: In modular arithmetic, we can add or subtract multiples of the modulus to both sides without changing the equivalence. Therefore, we can add 11 to both sides:
- x + 11 ≡ 2 + 11 (mod 11)
- x ≡ 13 (mod 11)
Step 3: Simplify 13 (mod 11) to its smallest non-negative residue:
13 ≡ 2 (mod 11)
So, the equation simplifies to:
- x ≡ 2 (mod 11)
Step 4: Since we have a negative value for x, we need to find its positive equivalent. We can add multiples of 11 to -x until we get a positive number. In this case, we can add 11:
-x + 11 ≡ 2 + 11 (mod 11)
11 - x ≡ 13 (mod 11)
Step 5: Simplify both sides:
11 - x ≡ 2 (mod 11)
- x ≡ 2 (mod 11)
Step 6: Multiply both sides by -1 to get a positive value for x:
x ≡ -2 (mod 11)
Step 7: Add multiples of 11 to -2 until we get a positive number. In this case, we can add 11:
x ≡ -2 + 11 (mod 11)
x ≡ 9 (mod 11)
So, the value of x in the set {0, 1, 2, ... 10} that satisfies the equation 3-x ≡ 5 (mod 11) is x ≡ 9 (mod 11).
To find the value of x that satisfies the equation 3-x ≡ 5 (mod 11), we can simplify it as follows:
1. Subtract both sides of the equation by 3:
(3-x) - 3 ≡ 5 - 3 (mod 11)
-x ≡ 2 (mod 11)
2. To remove the negative sign, multiply both sides by -1:
(-1)(-x) ≡ (-1)(2) (mod 11)
x ≡ -2 (mod 11)
3. Since we're working with the set {0, 1, 2, ... 10}, we need to find a number that is equivalent to -2 (mod 11) within this set.
4. To do this, we can keep adding or subtracting 11 from -2 until we get a number within the set:
-2 + 11 ≡ 9 (mod 11)
Therefore, the value of x that satisfies the equation 3-x ≡ 5 (mod 11) in the set {0, 1, 2, ... 10} is x = 9.