How many prime factors does positive integer n have?

Excellent Question.
Here is what i did in this one =
n=3*5 and 5^3 satisfy both the equations
Hence n can have one or two prime factors.

Hence E

I understand perfectly why (1) and (2) separately are insufficient, but I'm stuck at analyzing both statements together. Can anyone shed some light?

Hi Ed_Palencia

Statement 1 => implies, n can have two factors eg \((3 * 5)\)
\(or\)
can have only one prime factor, but \(n = 5 * 5\), so that \(n/5\) has one prime factor => insuff

Statement 2 => implies, \(3n^2\)=> two diff prime factors, so \(n\) can be \((3 * 5) or (5 * 5)\)
if \(n = 3 * 5\),
\(3n^2\) => \(3(3 * 5)\) => two prime factors
if \(n = 5 * 5\),
\(3n^2\) => \(3(5 * 5)\) => two prime factors

so if you combine, still n can be \(3 * 5 or 5 * 5\), not sufficient to tell how many prime factors

Answer (E)

Hope that helps !

shrouded1

For (1) isn't the statement saying that n/5 has only 1 prime factor? i.e: how can our tests show that it has 1 or 2 prime factors if statement says n/5 has only 1 prime factor?

Hi guys, I haven't understood why 1) is insufficient. My thought is:
if n/5 has only a prime factor, n/5 must be equal to 1,thus n = 5.

if n were 25(or any 5^x), n/5 would have 2 prime factors : 5 and 1.

Could you explain what I've missed?

1 is NOT a prime number.

2. Properties of Integers

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Hope it helps.

VERY nice one!

\(n \ge 1\,\,{\mathop{\rm int}}\)

\(?\,\, = \,\,\# \,\,{\rm{prime}}\,\,{\rm{factors}}\,\,{\rm{of}}\,\,n\)

\(\left( 1 \right)\,\,\,\,\left\{ \matrix{\\
\,{\rm{Take}}\,\,n = {5^2}\,\,\,\, \Rightarrow \,\,\,\,\,? = 1\,\,\,\,\,\,\,\,\,\,\left( {{\rm{just}}\,\,5} \right)\, \hfill \cr \\
\,{\rm{Take}}\,\,n = 5 \cdot 2\,\,\,\, \Rightarrow \,\,\,\,\,? = 2\,\,\,\,\,\,\,\left( {2\,\,{\rm{and}}\,\,5} \right) \hfill \cr} \right.\)

\(\left( 2 \right)\,\,\,\,\left\{ \matrix{\\
\,({\rm{Re}})\,{\rm{Take}}\,\,n = {5^2}\,\,\,\left( {3 \cdot {5^4}\,\,{\rm{has}}\,\,{\rm{just}}\,\,3\,\,{\rm{and}}\,\,5} \right)\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,? = 1\,\,\,\,\left( {{\rm{just}}\,\,5} \right)\,\,\,\,\,\,\,\,\, \hfill \cr \\
\,{\rm{Take}}\,\,n = 2 \cdot 3\,\,\,\left( {{3^3} \cdot {2^2}\,\,{\rm{has}}\,\,{\rm{just}}\,\,3\,\,{\rm{and}}\,\,2} \right)\,\,\,\,\,\, \Rightarrow \,\,\,\,\,? = 2\,\,\,\,\,\left( {2\,\,{\rm{and}}\,\,3} \right)\,\,\,\,\,\,\, \hfill \cr} \right.\)

\(\left( {1 + 2} \right)\,\,\,\left\{ \matrix{\\
\,({\rm{Re}})\,{\rm{Take}}\,\,n = {5^2}\,\,\,\left( {3 \cdot {5^4}\,\,{\rm{has}}\,\,{\rm{just}}\,\,3\,\,{\rm{and}}\,\,5} \right)\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,? = 1\,\,\,\,\left( {{\rm{just}}\,\,5} \right)\,\,\,\,\,\,\,\,\, \hfill \cr \\
\,{\rm{Take}}\,\,n = 3 \cdot 5\,\,\,\left( {{3^3} \cdot {5^2}\,\,{\rm{has}}\,\,{\rm{just}}\,\,3\,\,{\rm{and}}\,\,5} \right)\,\,\,\,\,\, \Rightarrow \,\,\,\,\,? = 2\,\,\,\,\,\left( {3\,\,{\rm{and}}\,\,5} \right)\,\,\,\,\,\,\, \hfill \cr} \right.\)


This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.

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