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If \(p,q,r,s\) are distinct integers such that:

\(\begin{array}{l} f\left( {p,q,r,s} \right) = max\left( {p,q,r,s} \right)\\ g\left( {p,q,r,s,} \right) = min\;\left( {p,q,r,s} \right)\\ h\left( {p,q,r,s} \right) = remainder\;of\frac{{p \times q}}{{r \times s}}if\;\left( {p \times q} \right) > \left( {r \times s} \right)\\ \;\left( {or} \right)remainder\;of\frac{{r \times s}}{{p \times q}}\;if\;\left( {r \times s} \right) > \left( {p \times q} \right) \end{array}\)

Also a function \(fgh\;\left( {p,q,r,s} \right) = f\left( {p,q,r,s} \right) \times g\left( {p,q,r,s} \right) \times h\left( {p,q,r,s} \right)\)

Also the same operations are valid with two variable function of the form \(f\left( {p,q} \right)\).

What is the value of \(fg\left( {h\left( {2,5,7,3} \right),4,6,8} \right)\)?

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Here, we have to determine the value of \(fg\left( {h\left( {2,5,7,3} \right),4,6,8} \right)\).

Given,

\(\begin{array}{l} h\left( {p,q,r,s} \right) = remainder\;of\frac{{p \times q}}{{r \times s}}if\;\left( {p \times q} \right) > \left( {r \times s} \right)\\ \;\left( {or} \right)remainder\;of\frac{{r \times s}}{{p \times q}}\;if\;\left( {r \times s} \right) > \left( {p \times q} \right) \end{array}\)

So, we obtain \(h\left( {2,5,7,3} \right) = remainder\;\left( {\frac{{7 \times 3}}{{2 \times 5}}} \right) = 1\)

Therefore, we get

\(fg\left( {h\left( {2,5,7,3} \right),4,6,8} \right) = fg\left( {1,4,6,8} \right)\)

Again, we have

\(\begin{array}{l} f\left( {p,q,r,s} \right) = max\left( {p,q,r,s} \right)\\ g\left( {p,q,r,s} \right) = min\left( {p,q,r,s} \right) \end{array}\)

So, we obtain

\(\begin{array}{l} fg\left( {1,4,6,8} \right) = f\;\left( {1,4,6,8} \right) \times g\left( {1,4,6,8} \right)\\ = max\left( {1,4,6,8} \right) \times min\left( {1,4,6,8} \right)\\ = 8 \times 1\\ = 8 \end{array}\)

Hence, the value of desired function is

\(fg\left( {h\left( {2,5,7,3} \right),4,6,8} \right) = fg\left( {1,4,6,8} \right) = 8\)

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