I’ve typed the question, in whose context my doubt is, and it’s answer at the end.
Please note that I do not require the solution as I’ve already understood how to find the answer via the given as well as other methods.
My actual question is:
$ 4(p+1)=7q $ and $(p+1)/7=q/4$ are essentially the same equations.
So why don’t we directly do $4(p+1)=7q=k$ , here k is the proportionality constant right?
Why do we get a wrong answer if we use any other form of the equation $(p+1)/7=q/4$ while equating it with k?
For ex:
$4(p+1)=7q$
$4(p+1)/7=q=k$
$p=(7k/4)-1$ and $q=k$
$(7k/4)-1≤102$ and $k≤102$ as p,q≤102
$k≤58.86$ and $k≤102$
$=>k≤58$
which is obviously wrong.
Question
Find the number of terms common to the two AP’s: 3,7,11…407 and 2,9,16,..709.
Answer
Let number of terms of two AP’s be m and n respectively.
$ 407=3+(m-1)*4$ and $709=2+(n-1)*7$
$=> m=102$ and $n=102 $
Let pth term of first AP and qth term of second AP be identical.
$3+(p-1)*4=2+(q-1)*7$
$4p-1=7q-5$
$4(p+1)=7q$
$(p+1)/7=q/4=k(say)$
$=> p=7k-1 and q=4k $
As max no. of terms for both AP’s is 102, p,q≤102.
$=>7k-1≤102$ and $4k≤102$
$=>k≤14.71$ and $k≤25.5$
$=> k≤14$ and for each value of k there exists a pair of identical terms. Hence, there are 14 identical terms.
Hence, we require the number of integer solutions, with $$1\leq \;p,q\;\leq102,\tag1$$ of the Diophantine equation $$7q-4p=4.\tag{*}$$
By observation†, $(p,q)=(6,4)$ is a particular solution. Thus, by this theorem, each solution is given by $$(p,q)=(6-7k,4-4k)\quad\text{for some }k\in\mathbb Z.\tag2$$ Think of $k$ as a counter for the solutions.
Solving $(1)$ and $(2)$ gives $$-13.71\leq k\leq0.71.$$ Thus, there are $14$ integer solutions.
†Let $n\in\mathbb N$ and $a,b\in\mathbb Z.$ Then, if (and only if) $b$ is a multiple of $\gcd(a,n),$ $$ax+ny=b$$ has an integer solution $(x,y),$ which can be obtained using the Euclidean algorithm.