A part of modular arithmetic I jut dont get, badly stuck for an answer. Help please?

Question

A housewife is travelling to market with all her eggs in one basket. She has between 100 and 200 eggs in the basket. Counting in threes there are 2 eggs leftover, counting in fives there are 2 eggs leftover and counting in sevens there are 3 eggs leftover. How many eggs are in the basket?

Answer 1

Could you write this up? $$\begin{align} x&\equiv 2 \pmod{3} \\ x &\equiv 2 \pmod{5} \\ x & \equiv 3 \pmod{7} \end{align}$$

These are small numbers, you can also find manually a common remainder of these three. In general, if $\gcd(m,n)=1$, then

$$\big( x\equiv a\pmod{m}\ \text{ and }\ x\equiv a\pmod{n} \big) \iff x\equiv a \pmod{mn}$$

Well, two of the remainders above are already common, so we can write one instead of them: $$\begin{align} x&\equiv 2 \pmod{15} \\ x & \equiv 3 \pmod{7} \end{align}$$

Now. Can you find a common remainder of them?

Well, for example $2\equiv 17 \pmod{15}$ and $17\equiv 10 \equiv 3 \pmod{7}$, so $x\equiv 17$ is good: $$x\equiv 17 \pmod{3\cdot 5\cdot 7}$$ How many solutions does it give between $100$ and $200$?

Answer 2

$x=2+3k$. Therefore $2+3k \equiv 2 \ \ (5)$. This gives $k=5t$. So, $x=2+15t$. Finally $2+15 t \equiv 3 \ \ (7)$ which means that $t=1+7s$. So, $x=17 +105 s$. For $s=1$ we get $x=122$.

Answer 3

Hint $\rm\ 5,7\:|\:2x\!+\!1\:\Rightarrow\:35\:|\:2x\!+\!1\:\Rightarrow\:mod\ 35\!:\ x\equiv \dfrac{-1}2\equiv \dfrac{34}2\equiv 17.\:$ Also $17\equiv 2\pmod 3.\:$

Answer 4

Use the Chinese remainder theorem. Working with the congruences $x \equiv 2 \pmod{3}$ and $x \equiv 2 \pmod{5}$, we compute the multiplicative inverse of 3 modulo 5 as 2, and of 5 modulo 3 as 2; hence $x \equiv 2 \cdot 5 \cdot 2 + 2 \cdot 3 \cdot 2 \equiv 2 \pmod {15}$. Repeating for the congruences $x \equiv 2 \pmod{15}$ and $x \equiv 3 \pmod{7}$, we get the multiplicative inverse of 7 modulo 15 is 13 and of 15 modulo 7 is 1; hence $x \equiv 2 \cdot 7 \cdot 13 + 3 \cdot 15 \cdot 1 \equiv 17 \pmod{105}$. 122 is the only integer between 100 and 200 with remainder 17 modulo 105.

Answer 5

Let $n$ be the number of eggs in the basket. By hypothesis, both $3$ and $5$ are factors of $n-2$, so $15$ is a factor of $n-2$, and so $n=15j+2$ for some integer $j$. Since $100\le n\le 200$ and $j$ is an integer, then $7\le j\le 13$. At this point, you want to see which of these $j$ will give $n=15j+2$ the form $n=7m+3$ for some integer $m$. In other words, you want to find for which of these $j$ that $7$ is a factor of $n-3=15j-1$. Well, $n-3=14j+j-1$, and $7$ is always a factor of $14j$, so we really only need to choose our $j$ so that $7$ is a factor of $j-1$. The only possibility is $j=8$, whence $n=15j+2=122$.