Let A=1,2,3, ,100.

Question

Let $A={1,2,3,\ldots \ldots \ldots \ldots ,100}$. Let $R$ be a relation on $A$ defined by $(x,y)\in R$ if and only if $2x=3y$. Let ${R}_{1}$ be a symmetric relation on $A$ such that $R\subset {R}_{1}$ and the number of elements in ${R}_{1}$ is $n$. Then, the minimum value of $n$ is \_\_\_\_\_\_\_\_\_.

Answer: 66

Accepted Answer

$2x=3y \Rightarrow x=3k,\ y=2k$ for $k=1,2,\ldots$. Since $x\leq100 \Rightarrow 3k\leq100 \Rightarrow k\leq33$ (binding constraint, as $y=2k\leq100$ allows up to $k=50$). So:
$$R = \{(3,2),(6,4),(9,6),\ldots,(99,66)\},\qquad n(R)=33$$

Since $x\neq y$ for every pair, making $R$ symmetric requires adding all 33 reverse pairs $(y,x)$:
$n = 33+33 = 66$

Solutions · 1