#number properties

[pwnthesat]

If c and d are integers and c^2-d^2 is even, which of the following must be true ? (A) cd id odd (B) cd is even (C) c + d is odd (D) c + d is even (E) c-d is odd Note: I plug in with numbers but there are two right answers B and D :/

c^2 – d^2 factors to (c + d)(c – d). So if c^2 – d^2 is even, then (c + d)(c – d) is even. The only way a product of two integers is even is if at least one of them is even. So either c + d or c – d or both must be even.

Since c and d are themselves integers, c + d and c – d must either both be even or both be odd. And just up above, we proved that one or both must be even for c^2 – d^2 to be even. So we know that c + d is even and c – d is even. Answer choice D is correct.

[pwnthesat]

which of the following expressions must be positive for all values of x? C) x^4 + x^2 + 4 E) x^6 + x^4 + 4(X^2) why not E?

If x = 0, then the expression in (E) can be 0, which is not positive.

[pwnthesat]

let T be the set of all integers that can be written as x^2-2, where x is a positive integer. All of the following are T Except>> (-2, -1, 2, 7, 14 ) Why is -2 the answer, isn't the square of negative two is 4 and subtracting a two it would be an integer?

–2 is the answer because –2 is 0^2 – 2, but 0 is not a positive integer. (It is an integer, and it’s even, but it’s not positive.)

[pwnthesat]

Official Q: 14, 17,32, 50, 83, 134, ... Starting with the third number in the sequence of numbers above, each number is 1 more than the sum of the two numbers just before it. for example, 32 = 1 + (14 + 17). of the first 100 numbers in this sequence, How many are even numbers?

This is all about knowing what happens when you add even and odd numbers together.

even + even = evenodd + odd = evenodd + even = odd

So the third term is 32 (even) because it’s made up of 14 (even) + 17 (odd) + 1 (odd). 

If you forget about the numbers and just look at the pattern, it goes like this:

even, odd, even, even, odd, even, even, odd, even, even, odd, …

After the first term, the pattern goes: odd, even, even, odd, even, even, …

2/3 of the terms after the first term are even. So 2/3 of the terms from 2–100 (there are 99 of those) will be even. That’s 66 terms. Then you have the first term, which is even. Add that in and you have 67 even terms in the first 100.

[pwnthesat]

2012-2013 SAT Test Section 2 #20, please.

[Link to test and answer key.]

In order for jn to have a units digit of 9, the units digits of j and n must multiply to 9. The only ways that can happen are if j and n both end in 3, or if one ends in 9 and the other ends in 1.

Since we know j, k, and n are consecutive integers in that order, it must be the case that j ends in 9, k ends in 0, and n ends in 1.

[pwnthesat]

what is the smallest possible integer value pf x for which (x-8)(x+5)>0? Can I just factor and set each equal to zero? (A) -8 (b) -7 (c) -6 (d) -5 (e) -4

I think you reversed the inequality here. –8 is the smallest answer choice, and it produces a value greater than 0: (-16)(-3) = 48. But -10 would produce a positive value, too: (-18)(-5) = 90. Which is to say that (A) is the smallest ANSWER CHOICE that makes the inequality true, but smaller there is no smallest possible value of x that'll make that answer true. You could go all the way to –∞.

However, if you're looking for the smallest integer value of x for which (x - 8)(x + 5) < 0, then you've got an interesting question. :)

Think of it this way. If you're multiplying two things, and getting a negative result, then one of them has to be positive, and one of them has to be negative:

[negative] × [positive] = [negative]

(x - 8) is going to be negative for all those answer choices, so you really just need to focus on finding the smallest possible integer value for which (x + 5) is positive. 

When x = -4, (x + 5) = 1. That's your answer. Any smaller, and you'd be at 0, or negative. 

That's the mathy solution, of course. You could also very easily backsolve. Drop the answer choices into the inequality and evaluate! It takes like 10 seconds, and erases any uncertainty about the algebra.

(A) (-8 - 8)(-8 + 5) = 48(B) (-7 - 8)(-7 + 5) = 30(C) (-6 - 8)(-6 + 5) = 14(D) (-5 - 8)(-5 + 5) = 0(E)  (-4 - 8)(-4 + 5) = -12

[crackthegmat-blog]

Day 21

The review from the night before was still fresh in my mind, so decided to cover some more material.

Completed the Number Properties Basics Workshop

It covered the following areas:

- Basics/Definitions

- Fractions

- Order of Operations

- Quiz Score - 73% = Sigh, I have scored high % in this section on the Practice tests in the past as well as my CATs, so this was shocking!

Had to attend a workout/social event, so called it a night for studying.

Was a fun night - started off with some Zumba (That was my 2nd time trying it, probably my last - NOT a big fan!) It was ok but I don't like workout classes that barely get you warmed up - I guess that was a good thing, given I was still recovering and not completely healthy.

This was followed by some veggie sandwich samplers @ the gym and a chair massage (ahhh much needed) We were all hungry and there was some sort of social hour at this place called South in Birmingham - It's a nice, loungy type of bar. Been there few times but first time having dinner there - Loved my Veggie salad! Decided to celebrate my friend's bday, which I had missed the week before by ordering her the Chocolate cake (Im a big time Chocolate addict, so that worked out perfect :) It was sooo rich and amazing! Decided to continue bday celebration @ a bar that was around the corner where one of my DJ friends spins - good music, good times, the fact that Carlos Delfino and some other Bucks' players happen to be hanging out at the same spot didn't hurt either :)

(Day 21: Feb 2, 2012)

[crackthegmat-blog]

Day 20

Decided to spend some time on the Critical Reasoning area since the material reviewed in class on Monday was still fresh in my mind.

Completed the online Critical Reasoning Basics Workshop.

This covered the following areas:

- Finding Conclusions and Evidence

- Assumptions

- KAPLAN 4 step Method

 Which basically states:

1) Read the question stem and determine the type of question

2) Read the stimulus

3) Think critically about your answer.

4) Evaluate the answer choices.

Quiz Score - 90% = Ok, I think I get it?

Will only find out once I complete all practice tests.

Attended the next KAPLAN Session - Arithmetic and Number Properties

This is all basic to me but I have also made some silly mistakes in the past in this area…So, you have to make sure you are detail oriented and double check some of the more tedious algebraic expression solutions before proceeding (if time allots)

(Day 20: Feb 1, 2012)