Elementary Number Theory
Definition : "Elementary number theory is a branch of mathematics that deals with the properties and relationships of integers, including topics like divisibility, prime numbers, and congruences, without relying on advanced mathematical concepts."
MATH-206
Past Paper 2019 Solution
Solve the following short questions.
(i) Use Euclidean Algorithm to find a solution of the equation 56x+72 Y = 8.
Solution:- Using Euclidean Algorithm
72=56 +16
56=16 (3) +8
(6 = 8(2) +o
gcd (56,72) =8
This equation is solvable.
Now,
For particular solution
8 = 56-(6(3)
8=56-(72-56) (3)
8 = 56-720456 (3)
8 = 56(4)-72 (3)
Xo = 4; Yo = 3
The general solutions will
Be:
X = Xo +(b/d)t ; Y=Yo(a/d) t
X = 4+(72/8)t ;
Y=3-(56/8)t
X=4+9t ; Y=3-7t
(ii) Is there exists an integer a such that 15 divides 6a-1? Justify your answer.
Solution:- 15/6a-1
6a-1= 15b; bez
6a-156=1 -> (1)
15 = 6(2)+3
6 = 3(2)+0
Gcd (15, 6) = 3
As 3 does not divides 1
So, equation 1 has no solution.
Hence no
Such ‘a’ exists.
If a = b (modm) then show that aⁿbⁿ (modm)
For all n≥1.
Solution:-
For n=1
a = b (modm) which is given as true Suppose that
aⁿ = bⁿ (modm).
=> a・aⁿ = b.bⁿ(modm)
=>aⁿ+1=bⁿ+1(modm)
Hence result is true
by Induction.
(iv) Describe (without proof) what do the solution the about ax + by = c.
Solution :-
If gcd of a ξ divides “c’
then this equation has solution. If
Gcd of aξb does not divides ‘c’ then this equation has no solution. Means solution does not exists.
If Solution exists, then
X = xo+(b/d)t ; Y=yo-(a/d)t
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