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What is the value of [3][7][3][7] in Z9\mathbb{Z}_{9} ? Express your answer as a co...
Mar 18, 2024
What is the value of [3][7][3][7] in Z9\mathbb{Z}_{9} ? Express your answer as a congruence class [X][X] where 0≤X≤80 \leq X \leq 8. X= 数字  X=\text { 数字 }
Solution by Steps
step 1
Calculate the value of 373^7 in Z9\mathbb{Z}_{9}
step 2
Since we are working in Z9\mathbb{Z}_{9}, we need to find the remainder when 373^7 is divided by 9
step 3
Using the asksia-ll calculation list, we have the sequence of multiples of 9: 9, 18, 27, 36, 45, 54, 63, 72, 81, 90, ..
step 4
The asksia-ll calculator indicates that 37mod  93^7 \mod 9 is 0
step 5
Therefore, the value of [3][7][3][7] in Z9\mathbb{Z}_{9} is [0][0]
Answer
X=0X=0
Key Concept
Modular Arithmetic
Explanation
In modular arithmetic, when we calculate abmod  na^b \mod n, we are looking for the remainder when aba^b is divided by nn. In this case, 37mod  93^7 \mod 9 equals 0, which means 373^7 is a multiple of 9.
what is the value of [2] [3] in mod 9? express your answer as a congruence class [X] where 0<= X <= 8
Solution by Steps
step 1
To find the value of [2][3][2][3] in mod 9, we first calculate the product of 2 and 3
step 2
The product of 2 and 3 is 2×3=62 \times 3 = 6
step 3
We then find the congruence class of 6 modulo 9
step 4
Since 6 is less than 9 and non-negative, it is already within the required range 0≤X≤80 \leq X \leq 8
step 5
Therefore, the congruence class of [2][3][2][3] in mod 9 is [6][6]
Answer
[6][6]
Key Concept
Multiplication in Modular Arithmetic
Explanation
In modular arithmetic, the product of two congruence classes is found by multiplying the representatives and then taking the result modulo the modulus. If the product is within the range of 0 to the modulus minus one, it is the representative of the congruence class.
what is the value of [3] [7] in mod 9? express your answer as a congruence class [X] where 0<= X <= 8
Solution by Steps
step 1
Calculate the value of 373^7 in mod 9
step 2
Since 33=273^3 = 27 which is congruent to 0 mod 9, 37=(33)2⋅33^7 = (3^3)^2 \cdot 3 is also congruent to 0 mod 9
step 3
Therefore, 373^7 mod 9 is 0
Answer
[37]≡[0]mod  9[3^7] \equiv [0] \mod 9 where 0≤X≤80 \leq X \leq 8, so X=0X=0.
Key Concept
Modular Exponentiation
Explanation
When calculating powers in modular arithmetic, if a power of the base is congruent to 0 mod n, then all higher powers of the base will also be congruent to 0 mod n.
Determine the inverse of [20][20] in Z41\mathbb{Z}_{41}. Just enter the number for the inverse, without any brackets. [20]−1=[ 数字 ] [20]^{-1}=[\text { 数字 }]
Solution by Steps
step 1
To find the modular inverse of [20][20] in Z41\mathbb{Z}_{41}, we need to find an integer xx such that 20x≡1(mod41)20x \equiv 1 \pmod{41}
step 2
We can use the Extended Euclidean Algorithm to find such an xx
step 3
Applying the Extended Euclidean Algorithm, we find that x=2x = 2 satisfies the equation 20⋅2≡40≡−1(mod41)20 \cdot 2 \equiv 40 \equiv -1 \pmod{41}
step 4
Since we want 20x≡1(mod41)20x \equiv 1 \pmod{41}, we can add 4141 to −1-1 to get 40+41=8140 + 41 = 81
step 5
Now we have 20⋅2≡81≡1(mod41)20 \cdot 2 \equiv 81 \equiv 1 \pmod{41}, which means 22 is the modular inverse of 2020 in Z41\mathbb{Z}_{41}
Answer
2
Key Concept
Modular Inverse
Explanation
The modular inverse of a number aa modulo mm is the number bb such that ab≡1(modm)ab \equiv 1 \pmod{m}. It can be found using the Extended Euclidean Algorithm.
Determine the inverse of [20][20] in Z41\mathbb{Z}_{41}. Just enter the number for the inverse, without any brackets. [20]−1=[ 数字 ] [20]^{-1}=[\text { 数字 }]
Solution by Steps
step 1
To find the modular inverse of [20][20] in Z41\mathbb{Z}_{41}, we need to solve the congruence 20x≡1(mod41)20x \equiv 1 \pmod{41}
step 2
We can use the Extended Euclidean Algorithm to find integers xx and yy such that 20x+41y=120x + 41y = 1
step 3
Applying the Extended Euclidean Algorithm, we find that x=2x = 2 satisfies the equation 20(2)+41(−1)=120(2) + 41(-1) = 1
step 4
Therefore, the modular inverse of [20][20] in Z41\mathbb{Z}_{41} is [2][2]
Answer
2
Key Concept
Modular Inverse
Explanation
The modular inverse of an integer aa modulo mm is the integer xx such that ax≡1(modm)ax \equiv 1 \pmod{m}. It can be found using the Extended Euclidean Algorithm.
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