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In the group {1, 2,........,16} under the operation of multiplication modulo 17, the order of the element 3 is
- a)4
- b)8
- c)12
- d)16
Correct answer is option 'D'. Can you explain this answer?
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Verified Answer
In the group {1, 2,........,16} under the operation of multiplication ...
{1 , 2 , . . . , 16} are non-zero elements of 34 = 81 = 13(mod 17)
38 = 13 x 13(mod 17)
= 16(mod 17)
312 = 13 x 16(mod 17)
= 208(mod 17)
= 4(mod 17)
316 = 13 x 4(mod 17)
= 52(mod 17) =l(mod 17)
The order o f the element 3 is 16 i.e. o(3) = 16
38 = 13 x 13(mod 17)
= 16(mod 17)
312 = 13 x 16(mod 17)
= 208(mod 17)
= 4(mod 17)
316 = 13 x 4(mod 17)
= 52(mod 17) =l(mod 17)
The order o f the element 3 is 16 i.e. o(3) = 16
Most Upvoted Answer
In the group {1, 2,........,16} under the operation of multiplication ...
Explanation:
To determine the order of an element in a group, we need to find the smallest positive integer n such that the element raised to the power of n gives the identity element of the group. In this case, the identity element is 1.
Given group: {1, 2, 3, ..., 16}
Operation: multiplication modulo 17
Step 1: Calculate the powers of 3 modulo 17 until we find the identity element:
3^1 ≡ 3 (mod 17)
3^2 ≡ 9 (mod 17)
3^3 ≡ 10 (mod 17)
3^4 ≡ 13 (mod 17)
3^5 ≡ 5 (mod 17)
3^6 ≡ 15 (mod 17)
3^7 ≡ 11 (mod 17)
3^8 ≡ 16 (mod 17)
3^9 ≡ 14 (mod 17)
3^10 ≡ 8 (mod 17)
3^11 ≡ 7 (mod 17)
3^12 ≡ 4 (mod 17)
3^13 ≡ 12 (mod 17)
3^14 ≡ 2 (mod 17)
3^15 ≡ 6 (mod 17)
3^16 ≡ 1 (mod 17)
Step 2: We can see that 3^16 ≡ 1 (mod 17), which means that the order of 3 is 16. The order of an element in a group is the smallest positive integer n such that a^n ≡ 1 (mod m), where m is the modulus.
Therefore, the correct answer is option 'D' (16).
To determine the order of an element in a group, we need to find the smallest positive integer n such that the element raised to the power of n gives the identity element of the group. In this case, the identity element is 1.
Given group: {1, 2, 3, ..., 16}
Operation: multiplication modulo 17
Step 1: Calculate the powers of 3 modulo 17 until we find the identity element:
3^1 ≡ 3 (mod 17)
3^2 ≡ 9 (mod 17)
3^3 ≡ 10 (mod 17)
3^4 ≡ 13 (mod 17)
3^5 ≡ 5 (mod 17)
3^6 ≡ 15 (mod 17)
3^7 ≡ 11 (mod 17)
3^8 ≡ 16 (mod 17)
3^9 ≡ 14 (mod 17)
3^10 ≡ 8 (mod 17)
3^11 ≡ 7 (mod 17)
3^12 ≡ 4 (mod 17)
3^13 ≡ 12 (mod 17)
3^14 ≡ 2 (mod 17)
3^15 ≡ 6 (mod 17)
3^16 ≡ 1 (mod 17)
Step 2: We can see that 3^16 ≡ 1 (mod 17), which means that the order of 3 is 16. The order of an element in a group is the smallest positive integer n such that a^n ≡ 1 (mod m), where m is the modulus.
Therefore, the correct answer is option 'D' (16).
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In the group {1, 2,........,16} under the operation of multiplication modulo 17, the order of the element 3 isa)4b)8c)12 d)16Correct answer is option 'D'. Can you explain this answer?
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