statements: (a) 7152n +3. 25n-2. (b) 131 3"+2+ 42n+1. (c) 27 25n+1 + 5n+2. (d) 43 16"+2 1 72n+1
Angles in Circles
Angles within a circle are feasible to create with the help of different properties of the circle such as radii, tangents, and chords. The radius is the distance from the center of the circle to the circumference of the circle. A tangent is a line made perpendicular to the radius through its endpoint placed on the circle as well as the line drawn at right angles to a tangent across the point of contact when the circle passes through the center of the circle. The chord is a line segment with its endpoints on the circle. A secant line or secant is the infinite extension of the chord.
Arcs in Circles
A circular arc is the arc of a circle formed by two distinct points. It is a section or segment of the circumference of a circle. A straight line passing through the center connecting the two distinct ends of the arc is termed a semi-circular arc.
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ELEMENTARY NUMBER THEORY
6. For n > 1, use congruence theory to establish each of the following divisibilit.
statements:
(a) 7|52m +3. 25n–2.
(b) 13|3"+2 + 42n+1.
+ 5n+2
(c) 27|25n+1
(d) 4316"+2 + 72n+1.
7. For n > 1, show that
(-13)"+1 = (-13)" +(-13)"-1 (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 a² = 1 (mod 24).
9. If p is a prime satisfying n < p < 2n, show that
(:)
2n
= 0 (mod p)
10. If a1, a2,..., an is a complete set of residues modulo n and gcd(a, n) = 1, prove that
aa¡, 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, ..., 2° 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
c, c+a, c+ 2a, c+ 3a, ... , c+ (n – 1)a
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 ninɔ).
4. Give an example to show that a = b* (mod n) and k = j (mod n) need not imply that
a = b (mod n).
5. Establish that if a is an odd integer, then for any n > 1
20
= 1 (mod 2"+2)
[Hint: Proceed by induction on n.]
Use the theory of congruences to verify that
89|24- 1
97|248 – 1
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