isi-entrance None Q4

isi-entrance · India · subjective_collection Number Theory GCD, LCM, and Coprimality

Let $X = \{0,1,2,3,\ldots,99\}$. For $a, b$ in $X$, we define $a * b$ to be the remainder obtained by dividing the product $ab$ by 100. For example, $9 * 18 = 62$ and $7 * 5 = 35$. Let $x$ be an element in $X$. An element $y$ in $X$ is called the inverse of $x$ if $x * y = 1$. Find all elements of $X$ that have inverses and write down their inverses.

$x * y = 1 \Rightarrow xy = 100k + 1$ for $x \in \{0,1,2,\ldots,99\}$.
(1) For $x = 1$, $y = 1$. Inverse of 1 is 1.
(2) There is no integral multiple of 2, 4, 5, 6 having 1 at unit place, $\Rightarrow 2, 4, 5, 6$ have no inverse.
(3) For $x = 3$: $3y = 100k + 1$. The least $k$ satisfying is 2, i.e. $3y = 201$, $y = 67$. $\therefore$ 3 has only inverse $= 67$.
(4) For $x = 7$: $7y = 100k + 1$. The least $k$ satisfying is 3, i.e. $7y = 301$, $y = 43$. $\therefore$ 7 has only inverse $= 43$.

Let $X = \{0,1,2,3,\ldots,99\}$. For $a, b$ in $X$, we define $a * b$ to be the remainder obtained by dividing the product $ab$ by 100. For example, $9 * 18 = 62$ and $7 * 5 = 35$. Let $x$ be an element in $X$. An element $y$ in $X$ is called the inverse of $x$ if $x * y = 1$. Find all elements of $X$ that have inverses and write down their inverses.

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