If the prime factorization of 784 is in the form of 2p-7q,then find th

1.	If the prime factorization of 784 is in the form of 2p-7q,then find the value of p+q
Answer» Appropriate Question :- If the prime factorization of 784 is in the form of $\sf \: {2}^{p} {7}^{q}, \: find \: the \: value \: of \: p + q$ $\large\underline{\sf{Solution-}}$ $\red{\bf :\longmapsto\:Prime \: factorization \: of \: 784}$ $\begin{gathered}\begin{gathered}\begin{gathered} \:\: \begin{array}{c\|c} {\underline{\sf{2}}}&{\underline{\sf{\:\:784\:\:\:}}}\\ {\underline{\sf{2}}}& \underline{\sf{\:\:392\:\:\:}} \\\underline{\sf{2}}&\underline{\sf{\:\:196\: \:\:}}\\{\underline{\sf{2}}}&{\underline{\sf{\:\:98\:\:\:}}} \\{\underline{\sf{7}}}&{\underline{\sf{\:\:49\:\:\:}}} \\ {\underline{\sf{7}}}&{\underline{\sf{\:\:7\:\:\:}}} \\ \underline{\sf{}}&{\sf{\:\:1\:\:\:}} \end{array}\end{gathered}\end{gathered}\end{gathered}$ $\red{\bf :\longmapsto\:Prime \: factorization \: of \: 784 = {2}^{4} \times {7}^{2} }$ It is given that, $\red{\bf :\longmapsto\:Prime \: factorization \: of \: 784 = {2}^{p} \times {7}^{q} }$ So, it implies, $\rm :\longmapsto\: {2}^{p} \times {7}^{q} = {2}^{4} \times {7}^{2}$ So, on COMPARING we get $\red{\rm :\longmapsto\:p = 4} \\ \red{\rm :\longmapsto\:q = 2}$ Therefore, $\bf :\longmapsto\:p + q = 4 + 2 = 6$ Additional Information :- Fundamental Theorem of Arithmetic :- This theorem STATES that : Every composite NUMBER can be FACTORIZED as a product of PRIMES and this factorization is unique irrespective of their places. Euclid Division Lemma :- Let a and b are positive integers such that a > b, then their exist integers q and r such that a = bq + r, where r belongs to [0, b).

If the prime factorization of 784 is in the form of 2p-7q,then find the value of p+q

Answer»

Appropriate Question :-

If the prime factorization of 784 is in the form of

$\sf \: {2}^{p} {7}^{q}, \: find \: the \: value \: of \: p + q$

$\large\underline{\sf{Solution-}}$

$\red{\bf :\longmapsto\:Prime \: factorization \: of \: 784}$

$\begin{gathered}\begin{gathered}\begin{gathered} \:\: \begin{array}{c|c} {\underline{\sf{2}}}&{\underline{\sf{\:\:784\:\:\:}}}\\ {\underline{\sf{2}}}& \underline{\sf{\:\:392\:\:\:}} \\\underline{\sf{2}}&\underline{\sf{\:\:196\: \:\:}}\\{\underline{\sf{2}}}&{\underline{\sf{\:\:98\:\:\:}}} \\{\underline{\sf{7}}}&{\underline{\sf{\:\:49\:\:\:}}} \\ {\underline{\sf{7}}}&{\underline{\sf{\:\:7\:\:\:}}} \\ \underline{\sf{}}&{\sf{\:\:1\:\:\:}} \end{array}\end{gathered}\end{gathered}\end{gathered}$

$\red{\bf :\longmapsto\:Prime \: factorization \: of \: 784 = {2}^{4} \times {7}^{2} }$

It is given that,

$\red{\bf :\longmapsto\:Prime \: factorization \: of \: 784 = {2}^{p} \times {7}^{q} }$

So, it implies,

$\rm :\longmapsto\: {2}^{p} \times {7}^{q} = {2}^{4} \times {7}^{2}$

So, on COMPARING we get

$\red{\rm :\longmapsto\:p = 4} \\ \red{\rm :\longmapsto\:q = 2}$

Therefore,

$\bf :\longmapsto\:p + q = 4 + 2 = 6$

Additional Information :-

Fundamental Theorem of Arithmetic :-

This theorem STATES that : Every composite NUMBER can be FACTORIZED as a product of PRIMES and this factorization is unique irrespective of their places.

Euclid Division Lemma :-

Let a and b are positive integers such that a > b, then their exist integers q and r such that a = bq + r, where r belongs to [0, b).