Let p, q, r and s be natural numbers such that p - 2016 = q + 2017 = r - 2018 = s + 2019 Which one of the following is the largest natural number?

Correct Option (3)

Let the common value of the given expression be $k$k.
$$p - 2016 = q + 2017 = r - 2018 = s + 2019 = k$$p−2016=q+2017=r−2018=s+2019=k
From this, each variable can be expressed in terms of $k$k:

• $p = k + 2016$p=k+2016
• $q = k - 2017$q=k−2017
• $r = k + 2018$r=k+2018
• $s = k - 2019$s=k−2019

To identify the largest natural number, we compare the constant terms associated with $k$k. The constants are $+2016$+2016, $-2017$−2017, $+2018$+2018, and $-2019$−2019. The largest among these constants is $+2018$+2018. Consequently, $r = k + 2018$r=k+2018 represents the largest natural number.

Incorrect Options:

Based on the expressions derived:

• $p = k + 2016$p=k+2016
• $q = k - 2017$q=k−2017
• $s = k - 2019$s=k−2019

Comparing these with $r = k + 2018$r=k+2018:

• Since $2016 < 2018$2016<2018, $p$p is less than $r$r.
• Since $-2017 < 2018$−2017<2018, $q$q is less than $r$r.
• Since $-2019 < 2018$−2019<2018, $s$s is less than $r$r.

The complete order from smallest to largest is $s < q < p < r$s<q<p<r. Therefore, $p, q,$p,q, and $s$s are not the largest natural numbers.