1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
kkurt [141]
3 years ago
11

What is the smallest integer $n$, greater than $1$, such that $n^{-1}\pmod{130}$ and $n^{-1}\pmod{231}$ are both defined?

Mathematics
1 answer:
olasank [31]3 years ago
3 0

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

\begin{cases} 17x \equiv 1 \pmod{130} \\ 17y \equiv 1 \pmod{231} \end{cases}

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.

You might be interested in
How to solve radical numerator fractions
Nonamiya [84]
Y=radical 3
just cancel out the 2 on both sides
3 0
3 years ago
I do not understand how to solve this. Please can someone explain in a way i can understand.
Hoochie [10]

This π (pi) means 22/7 (pi).
7 0
3 years ago
Swati recorded this data set, which contains an outlier.
podryga [215]

Answer:

lower quartile: 163  upper quartile: 184

Step-by-step explanation:

1. put the numbers in order from least to greatest. 97, 163, 169, 175, 184, 199

2. find the median. 172

3. find the median of the first 3 numbers to get your Q1. 163

4. find the median of the last 3 numbers to get your Q3. 184

3 0
3 years ago
Which of the following proportions could be used to convert 6 days to hours?
nekit [7.7K]

Answer:

Step-by-step explanation:

24 hours---------->1 day

x hours------------>6 days

Cross multiplying:

x=24x6

x=144 hrs

4 0
3 years ago
Chapter one course 2
Marysya12 [62]
What you need help with
8 0
3 years ago
Other questions:
  • Travis was attempting to make muffins to take to a neighbor. The recipe that he was using required 3/4 cup of sugar and 1/8 cup
    12·1 answer
  • What the answer for $9; 10% increase
    13·2 answers
  • At a sale on winter clothing, Cody bought two pairs of gloves and four hats for $43.00. Tori bought two pairs of gloves and two
    11·2 answers
  • The vertex of this parabola is at (4, -3). When the x-value is 5, the
    14·1 answer
  • Which of the following is a disadvantage of a personal
    15·1 answer
  • Train A departs station one traveling at 75 mph. Train N departs station two 30 minutes later traveling at 45 mph. The stations
    6·1 answer
  • Evaluate the expression a=7 and x = -6
    9·1 answer
  • What is 6*56577467577467475
    13·2 answers
  • You recently deposited $2000 in a bank account during your first year at work.
    6·1 answer
  • In which step did the veterinarian make the first error?<br> Step 1<br> Step 2<br> Step 3<br> Step 4
    11·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!