Is there a quick way to do this problem using number prop? (oops wrong forum. i can't move this. sorry)
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Hey Knicks,
Definitely! And I love that you're thinking that way...the more you're thinking through the Number Properties lens, the easier it is to save time and energy on quant problems.
When they ask "is x divisible by 2", you should really start thinking "what does it mean to be divisible by 2?". Well, that's an even number - the definition of "even" is essentially "divisible by 2". So with that in mind, your only options for the two statements (since we know that x is an integer) are "even", "odd", or "could be either". And we have some really defined rules for addition/subtraction and for multiplication when it comes to evens and odds:
Add/Subtract
Even + Even = Even
Odd + Odd = Even
Odd + Even = Odd
Multiply
Even * Even = Even
Even * Odd = Even
Odd * Odd = Odd
***Note: it's really important to know that these rules exist, but you probably don't need to memorize them...as long as you know they exist you can prove them to yourself with 1s and 2s really quickly.
So for this problem, statement 1 tells us that x^3 + x is divisible by 4. Well, that means that it's even. And we know that the two ways to add to an even number are:
Odd + Odd
or
Even + Even
And since x^3 and x will both be the same, as far as even/odd, we can have either of the above - if x is odd, x^3 will be odd, and we have the Odd + Odd = Even version. If x is even, we have the Even + Even version. So x could be either even or odd, and statement 1 is not sufficient.
Statement 2 says that 5x + 4 is divisible by 6, which means that it's also even. And, again, our only options to add to even are:
Odd + Odd ---> But we can't have that, since we're already stuck with an even + 4 by the statement, so we can't use this one.
Even + Even
This one we can use, but if x were odd, 5x would be odd (Odd*Odd=Odd), so in order for us to use this, x MUST BE even. Therefore, we can prove that x must be even, as that's the only type of number that allows us to satisfy the statement.
I think the biggest key here is to recognize up front that the question is asking "is x EVEN?". From there, "...divisible by 4"; "...divisible by 6" and statements like that are really only worthwhile to you as indications of even or odd, so you don't have to do the algebra...you can just test the number properties.
