Number of ordered pairs (x, y) of integers su | vedclass

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(C) Let the product $xy = n$,where $1 \le n \le 99$. Since $x$ and $y$ are integers and $xy > 0$,both $x$ and $y$ must have the same sign.Case $1$: $x

Vedclass · Accepted Answer

$946$

Vedclass · Answer

(C) Let the product $xy = n$,where $1 \le n \le 99$. Since $x$ and $y$ are integers and $xy > 0$,both $x$ and $y$ must have the same sign.Case $1$: $x, y > 0$. For a fixed $n$,the number of pairs $(x, y)$ is the number of divisors of $n$,denoted by $d(n)$. The total number of pairs is $\sum_{n=1}^{99} d(n)$.Using the property $\sum_{n=1}^{N} d(n) = \sum_{k=1}^{N} \lfloor \frac{N}{k} \rfloor$,for $N=99$:$\sum_{n=1}^{99} d(n) = \lfloor \frac{99}{1} \rfloor + \lfloor \frac{99}{2} \rfloor + \dots + \lfloor \frac{99}{99} \rfloor = 99 + 49 + 33 + 24 + 19 + 16 + 14 + 12 + 11 + 9 + 9 + 8 + 7 + 7 + 6 + 6 + 5 + 5 + 5 + 4 + 4 + 4 + 4 + 4 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 1 + 1 + \dots + 1 = 473$.Case $2$: $x, y  0$. Then $xy = ab = n$,where $1 \le n \le 99$. This is identical to Case $1$,so there are $473$ pairs.Total pairs = $473 + 473 = 946$.

Number of ordered pairs $(x, y)$ of integers such that their product is a positive integer less than $100$ is -

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