Number Properties

Question 1

What are solutions to the following in terms of evens and odds?

1) EVEN + EVEN = ?
2) ODD + EVEN = ?
3) ODD + ODD = ?

Answer 1

1) EVEN + EVEN = EVEN
2) ODD + EVEN = ODD
3) ODD + ODD = EVEN

Same goes for subtraction
EVEN - EVEN = EVEN
ODD - EVEN = ODD
ODD - ODD = EVEN

Question 2

If a + b = even, then (a)(b) = ?

Answer 2

You don’t know…

1) Even + Even = Even
BUT ALSO
2) Odd + Odd = Even

SO

1) a=2, b=2 (ab = 4)
2) a=3, b=3 (ab = 9)

Question 3

The greatest possible factor of a and b is…

Answer 3

• *The positive distance between them.**
  (e. g., |a-b| = X. 2 numbers that are X apart may both be multiples of X, but cannot both be multiples of any number larger than X.

IE… 63 and 70, GPF = 7
NOT both multiples of 8, 9, or 10

BUT here’s where the may comes in
a = 21 & b = 7 … 14 is the greatest possible factor, but it’s actually not a factor of 7, SO the GPF here is 7.

Question 4

The GMAT will throw you 3 types of Number Properties problems. How can you tell that the GMAT is testing

1) Evens & Odds
2) Divisibility & Primes
3) Positives & Negatives

Answer 4

Question 5

If something in the Q text says “A number divided by 2” what type of problem is this testing?

Answer 5

Evens and Odds!
Anything divided by 2 is Even.

Question 6

What is this question testing?

In a certain game, a machine contains tickets worth 2, 3, 5, and 7 points. Each time a person plays, a number of tickets are dispensed, and the score is calculated as the product of the point values on each ticket…

Answer 6

Divisibility and Primes!

Notice that the values given are all prime numbers, and the question mentions the word product.

HINT: A product of prime numbers is giving the hint to think about factors and factoring

Question 7

What type of problem is this testing?

3s - 2t is even, where s and t are integers.

Answer 7

Evens and Odds!
For 3s - 2t to be even, then it MUST BE (O - O) or (E - E). 2t must be even (but we can’t say was t is), so 3s must be even, so s must be even.

Question 8

If you see this in a question, what is it testing / what does it tell you?

X/Y > 1

Answer 8

Positive and Negatives!

1) Since the quotient is positive, they must both have the same sign
2) Since the quotient > 1, it must be an improper fraction (X > Y)

Question 9

What is the following testing?

n is an integer, but (100+n)/n is not an integer…

Answer 9

Divisibility and Primes!

-Divide the n into the numerator and you get (100/n) + 1
SO
you know that n is not a factor of 100

Question 10

What is this testing:
n has exactly 2 positive factors.

Answer 10

Divisibility and Primes!

n must be a prime #!

Question 11

What is this telling us?

Integer M has an odd number of factors.

Answer 11

M is a pefect square!

Only perfect squares have an odd number of factors.

Question 12

What is this telling us?

If X = 27! + 5, then X is divisible by which of the following?
(think 27 x 26 x 25 … 5 x 4 x 3…)

AND if 2 numbers are divisible by a number, what is true about their SUM?

Answer 12

X is divisible by 5!

RULE: If 2 numbers are divisible by a number, their SUM is also divisible by that number.

27! is divis. by 5 and so is 5! (therefore 27! + 5 is divis. by 5)

Question 13

3^k is a factor of P –> what is this actually saying?

Answer 13

First off, P is a multiple of 3.

Second off, ask yourself “How many 3s go into P”?

3^k = 3x3x3x3x3….

A certain number of 3s go into P. You’re just trying to find how many 3s there are

Question 14

Q: If p is the product of the integers from 1 to 30, inclusive, what is the greatest integer k for which 3^k is a factor of p?

• 1) What is this really asking?
  2) What is the key here?*

Answer 14

1) This is really asking “How many 3s go into p?”
2) the KEY is writing out a list of the multiples of 3 that go into 30!
3, 6, 9, 12, 15, 18, 21, 24, 27, 30

NOW, if you factor all these out, you can see that there are 14 3’s in p.

Question 15

X + Y > 0

What is this saying?

Answer 15

At least one of these is positive!

Question 16

X + Y < 0

What is this saying?

Answer 16

At least 1 of these is negative!

Question 17

Y^x < 0

What is this saying about X and Y?

Answer 17

Well…

Y MUST be negative (if it’s positive, Y^x will always be greater than 0)

X must be ODD, but CAN be EITHER positive or negative

Question 18

If positive integer X is a multiple of 6 and positive integer y is a multiple of 14, is xy a multiple of 105?

• 1) When given “X is a multiple of 6” - what is the first step?
  2) What is this question really asking?*

Answer 18

1) Create a factor tree with X on top! That is what you should always do when you see “something is a multiple of something else”!
2) SO, if xy were to be a multiple of 105, you’d make a factor tree with xy on top and 105 below…THEN you’d realize that the only prime factor missing (from the factor trees of x and y) is 5

Question is asking: Is xy a multiple of 5?

Question 19

How do you know a number is divisible by 9?

Answer 19

The sum of its digits is divisible by 9!

IE…(819 –> 8+1+9 = 18, 18 is divis. by 9, so YES!)

819/9 = 91

Question 20

What is true about (Even) x (Even) #s and their relationship to 4?

Answer 20

An even * another even = divisible by 4! (6)(8) = 48 [48/4=12] (12)(12) = 144 [144/4=36]

Question 21

How do you know if a number is divisible by 6?
AND What is true about any (Even)(3)?

Answer 21

1) It is EVEN and Divisible by 3 (sum of its integers is divis. by 3)

THEREFORE, any (Even)(3) = Divis. by 6

(12) (3) = 36 [36/6 = 6]
(24) (3)=72 [72/6=12]