Question : Let $p, q, r$ and $s$ be positive natural numbers having three exact factors including 1 and the number itself. If $q>p$ and both are two-digit numbers, and $r>s$ and both are one-digit numbers, then the value of the expression $\frac{p-q-1}{r-s}$ is:
Option 1: – s –1
Option 2: s – 1
Option 3: 1 – s
Option 4: s + 1
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Correct Answer: – s –1
Solution :
Given: $q > p$ and both are two-digit numbers
So, possible values = $5^2$ and $7^2$
According to the concept,
$p = 5^2 = 25$ [Factors = 1, 5, 25]
$q = 7^2 = 49$ [Factors = 1, 7, 49]
Here, $r > s$ and both are one-digit numbers
So, possible values = $2^2$ and $3^2$
$s = 2^2 = 4$ [Factors = 1, 2, 4]
$r = 3^2 = 9$ [Factors = 1, 3, 9]
$\therefore \frac{p-q-1}{r-s}$
$= \frac{25-49-1}{9-4}$
$=\frac{- 25}{5}$
= – 5
$- 5 = – 4 – 1 = –s – 1$
Hence, the correct answer is $–s – 1$.
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