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NEED HELP IN CS701 ASSIGNMENT NO.1
DUE DATE 09-11-2017
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Question No 2 required
if some solve,
then
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The given equation 33x + 15y = 14 has no integral solutions.
I’d prove a more generic and stronger claim.
Claim : The equation ax+by=c
has integers if and only if gcd(a,b)|c.
Observe that it works both ways.
Proof :
For the forward direction.
Given ax+by=c
has integer solution. To prove gcd(a,b)|c.
Assume gcd(a,b)=k.
∴a=kq
∴b=kr
for some q,r∈I
∴kqx+kry=c
∴k(qx+ry)=c
∴k|c⇒gcd(a,b)|c.
Hence the forward direction proof is complete.
For the reverse direction.
Given gcd(a,b)|c.
To prove ax+by=c
has integer solution.
Since k
is the gcd(a,b) there exist integers x′,y′∈Z such that ax′+by′=k.
Also, k|c⇒c=kd
for some integer d.
∴ax′+by′=k
∴d(ax′+by′)=kd
∴a(dx′)+b(dy′)=c
Implies that ax+by=c
has integer solution.
This proves our claim.
for given equation, gcd(15,33)=3∤14.
Hence no integer solution.
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Question No 2
CS701 Assignment Qustion No 1 Solution
Sol.
Ax+By=C
Coefficient ko comapre karain with given equation 33x+15y=14
A=33
B=15
C=14
The greatest common factor (GCR) of A and B must be divisible by C
GCF of A and B is 3
33/3=11
And
15/3=5
But this 3 is not not divisible by 14
As
14/3===not divisible
hence the given statement has no integer solution
check video also
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