[Hint: Proceed by induction on n.] 16. Use the theory of congruences to verify that 89| 24 – 1 and 97| 248 – 1 - 17. Prove that whenever ab = cd (mod n) and b d (mod n), with ged(b, n) =
Correlation
Correlation defines a relationship between two independent variables. It tells the degree to which variables move in relation to each other. When two sets of data are related to each other, there is a correlation between them.
Linear Correlation
A correlation is used to determine the relationships between numerical and categorical variables. In other words, it is an indicator of how things are connected to one another. The correlation analysis is the study of how variables are related.
Regression Analysis
Regression analysis is a statistical method in which it estimates the relationship between a dependent variable and one or more independent variable. In simple terms dependent variable is called as outcome variable and independent variable is called as predictors. Regression analysis is one of the methods to find the trends in data. The independent variable used in Regression analysis is named Predictor variable. It offers data of an associated dependent variable regarding a particular outcome.
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ELEMENTARY NUMBER THEORY
n2 1, use congruence theory to establish each of the following divisibility
ements:
7152+3.25-2.
1313+2+42n+1.
(c) 2712 5m+1 + 5n+2_
(d) 4316*+2 + 72n+1.
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7. For n 2 1, show that
(-13)"+ = (-13)" +(-13)"- (mod 181)
[Hint: Notice that (-13)² = -13+1 (mod 181); use induction on n.]
8. Prove the assertions below:
(a) If a is an odd integer, then a? = 1 (mod 8).
(b) For any integer a, a³ = 0, 1, or 6 (mod 7).
(c) For any integer a, a“ = 0 or 1 (mod 5).
(d) If the integer a is not divisible by 2 or 3, then a? = 1 (mod 24).
9. If p is a prime satisfying n < p < 2n, show that
whe
the
If
2n
= 0 (mod p)
N
10. If a1, a2, ..., an is a complete set of residues modulon and gcd(a, n) = 1, prove that
aaj, aa2, . , aa, is also a complete set of residues modulo n.
[Hint: It suffices to show that the numbers in question are incongruent modulo n.]
11. Verify that 0, 1, 2, 2², 2³, ..., 2º form a complete set of residues modulo 11, but that
0, 12, 22, 3², .., 10² do not.
12. Prove the following statements:
(a) If gcd(a, n) = 1, then the integers
с, с +а,с +2а, с + За, ..., с + (п -1)а
form a complete set of residues modulo n for any c.
(b) Any n consecutive integers form a complete set of residues modulo n.
[Hint: Use part (a).]
(c) The product of any set of n consecutive integers is divisible by n.
13. Verify that if a = b (mod n¡) and a = b (mod n2), then a = b (mod n), where the integer
n = lcm(n1, n2). Hence, whenever n¡ and n2 are relatively prime, a = b (mod n¡n2).
14. Give an example to show that ak = bk (mod n) and k = j (mod n) need not imply that
a' = b (mod n).
15. Establish that if a is an odd integer, then for any n > 1
a" = 1 (mod 2"+2)
[Hint: Proceed by induction on n.]
16. Use the theory of congruences to verify that
89| 244 - 1
and
97 |248 – 1
17. Prove that whenever ab = cd (mod n) and
a =c (mod n).
18. If a =b (mod n1) and a =C (mod n2), prove that b c (mod n), where the integer
=d (mod n), with ged(b, n) = 1, then
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n gcd(n1, n2).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3bf7da54-5138-4b57-a0f9-aec30fa8e8cf%2F41356b0c-b5c4-431d-a51f-f8cf50c1dc0d%2F3mvnzdf_processed.jpeg&w=3840&q=75)
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