Write 100 as the sum of two positive integers

From Bezout's Lemma, note that since $\gcd(7,11) = 1$, which divides $100$, there exists $x,y \in \mathbb{Z}$ such that $7x+11y=100$.

A candidate solution is $(x,y) = (8,4)$.

The rest of the solution is given by $(x,y) = (8+11m,4-7m)$, where $m \in \mathbb{Z}$. Since we are looking for positive integers as solutions, we need $8+11m > 0$ and $4-7m>0$, which gives us $-\frac8{11}<m<\frac47$. This means the only value of $m$, when we restrict $x,y$ to positive integers is $m=0$, which gives us $(x,y) = (8,4)$ as the only solution in positive integers.

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If you do not like to guess your candidate solution, a more algorithmic procedure is using Euclid' algorithm to obtain solution to $7a+11b=1$, which is as follows.

We have \begin{align} 11 & = 7 \cdot (1) + 4 \implies 4 = 11 - 7 \cdot (1)\\ 7 & = 4 \cdot (1) + 3 \implies 3 = 7 - 4 \cdot (1) \implies 3 = 7 - (11-7\cdot (1))\cdot (1) = 2\cdot 7 - 11\\ 4 & = 3 \cdot (1) + 1 \implies 1 = 4 - 3 \cdot (1) \implies 1 = (11-7 \cdot(1)) - (2\cdot 7 - 11) \cdot 1 = 11 \cdot 2-7 \cdot 3 \end{align} This means the solution to $7a+11b=1$ using Euclid' algorithm is $(-3,2)$. Hence, the candidate solution $7x+11y=100$ is $(-300,200)$. Now all possible solutions are given by $(x,y) = (-300+11n,200-7n)$. Since we need $x$ and $y$ to be positive, we need $-300+11n > 0$ and $200-7n > 0$, which gives us $$\dfrac{300}{11} < n < \dfrac{200}7 \implies 27 \dfrac3{11} < n < 28 \dfrac47$$ The only integer in this range is $n=28$, which again gives $(x,y) = (8,4)$.

An effort to avoid any enumeration or hiding an inductive leap to the answer.

$7a + 11b = 100: a,b \in N$

$ 11b \leq 100 - 7 = 93$

$\implies 1 \leq b \leq 8$

$ 7(a+b) = 100 - 4b$

$\implies 100 - 4b \equiv 0 \mod 7$

$\implies 25 - b \equiv 0 \mod 7 $

$\implies b \equiv 25 \mod 7 $

$\implies b \equiv 4 \mod 7$

$ \implies b = 4 + 7n$

We know $ 1 \leq b \leq 8 $.

So we have $b \equiv 25 \mod 7$, so $ b = 4$ and hence $a = 8$.

But is it the smart way of doing it?

You are asked to find a and b so that $7a+11b=100\iff7(a+b)+4b=100\iff$

$4\Big[(a+b)+b\Big]+3(a+b)=100\iff3\Big\{\Big[(a+b)+b\Big]+(a+b)\Big\}+\Big[(a+b)+b\Big]=$ $100$.

But $100=99+1=3\cdot33+1$, so $a+2b=1\iff2a+4b=2$, and $2a+3b=33$. It follows
that $b_0=-31$ and $a_0=63$ is one solution. But, then again, so are all numbers of the form $a_k=63-11k$ and $b_k=-31+7k$, with $k\in$ Z. All we have to do now is pick one pair whose components are both positive. The first equation implies $k<6$, and the latter $k>4$.