Ronald wanted an allowance. his father gave him a choice of getting it on a weekly or on a daily basis. He said he would either
2 answers:
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First of all, the modular inverse of n modulo k can only exist if GCD(n, k) = 1.
We have
130 = 2 • 5 • 13
231 = 3 • 7 • 11
so n must be free of 2, 3, 5, 7, 11, and 13, which are the first six primes. It follows that n = 17 must the least integer that satisfies the conditions.
To verify the claim, we try to solve the system of congruences
Use the Euclidean algorithm to express 1 as a linear combination of 130 and 17:
130 = 7 • 17 + 11
17 = 1 • 11 + 6
11 = 1 • 6 + 5
6 = 1 • 5 + 1
⇒ 1 = 23 • 17 - 3 • 130
Then
23 • 17 - 3 • 130 ≡ 23 • 17 ≡ 1 (mod 130)
so that x = 23.
Repeat for 231 and 17:
231 = 13 • 17 + 10
17 = 1 • 10 + 7
10 = 1 • 7 + 3
7 = 2 • 3 + 1
⇒ 1 = 68 • 17 - 5 • 231
Then
68 • 17 - 5 • 231 ≡ = 68 • 17 ≡ 1 (mod 231)
so that y = 68.
Answer:
<em>The denominator is 12</em>
Step-by-step explanation:
<u>The Number Line</u>
We can represent any real number in a properly segmented line, where the zero-mark is the separation between positive numbers to its right side and negative numbers to the left side.
We are given the number x = 2/3. Since it's positive, its location lies to the right of the zero but before the 1 as shown in the figure below.
We want to locate a second point y at the same distance from 0 as the point x, so we have two options: y is the same as x, or y is negative at the same distance from zero. We chose the last option.
Its numerator must be 8, so we need to multiply it by 4. But we cannot alter the value of x, so we must also multiply the denominator by 4, which gives us the fraction 8/12, but recall it must be negative.
The denominator is 12
