Let m and $\mathrm{n},(\mathrm{m} \lt \mathrm{n})$ be two 2-digit numbers. Then the total numbers of pairs $(m, n)$, such that $\operatorname{gcd}(m, n)=6$, is ________

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Aicharya · Accepted Answer

Let $\mathrm{m}=6 \mathrm{a}, \mathrm{n}=6 \mathrm{~b}$
$\mathrm{m} \lt \mathrm{n} \Rightarrow \mathrm{a} \lt \mathrm{b}$
where $\mathrm{a} \& \mathrm{~b}$ are co-prime numbers also since $m \& n$ are 2 digit nos, so
$10 \leq \mathrm{m} \leq 99 \& 10 \leq \mathrm{n} \leq 99$
i.e. $2 \leq \mathrm{a} \leq 16 \& 2 \leq \mathrm{b} \leq 16$
( $\because$ a is integer)
Now
$2 \leq \mathrm{a} \lt \mathrm{b} \leq 16 \& \quad \mathrm{a} \& \mathrm{~b}$ are co-prime
$\therefore$ if
$\mathrm{a}=2, \mathrm{~b}=3,5,7,9,11,13,15$
$a=3, b=4,5,7,8,10,11,13,14,16$
$a=4, b=5,7,9,11,13,15$
$a=5, b=6,7,8,9,11,12,13,14,16$
$\mathrm{a}=6, \mathrm{~b}=7,11,13$
$\mathrm{a}=7, \mathrm{~b}=8,9,10,11,12,13,15,16$
$\mathrm{a}=8, \mathrm{~b}=9,11,13,15$
$a=9, b=10,11,13,14,16$
$\mathrm{a}=10, \mathrm{~b}=11,13$
$\mathrm{a}=11, \mathrm{~b}=12,13,14,15,16$
$\mathrm{a}=12, \mathrm{~b}=13$
$a=13, b=14,15,16$
$\mathrm{a}=14, \mathrm{~b}=15$
$\mathrm{a}=15, \mathrm{~b}=16$
64 ordered pairs

Let m and $\mathrm{n},(\mathrm{m} \lt \mathrm{n})$ be two 2-digit numbers. Then the total numbers of pairs…

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