Number Properties with Profit: GMAT, GRE QUANT

If the selling price of an item is doubled then the profit is tripled. What is the percentage of profit?


A) 50%
B) 100%
C) 150%
D) 200%
E) 250%

Overlapping Sets In An Interesting GMAT, GRE Quant Problem

There are 750 male and female participants in a meeting. Half of the female participants and one-quarter of the male participants are Democrats. One-third of all the participants are Democrats. How many of the Democrats are female?

(A) 75
(B) 100
(C) 125
(D) 175
(E) 225

Relative Range Distance Rate Time GMAT, GRE Problem

The distance between Mercury and Earth changes due to the orbits of the planets. When Mercury is at its closest point to Earth, it is 48 million miles away. When Mercury is at its furthest point from Earth, it is 138 million miles away. For a science project, Ruby calculates the maximum and minimum amount of time it would take to travel from Earth to Mercury in a spacecraft traveling 55 miles per hour. Approximately what are the times, in days?

(A) 3,636 and 10,454
(B) 14,545 and 41,818
(C) 36,364 and 104,545
(D) 87,272 and 250,909
(E) 872,727 and 2,509,091

GMAT, GRE Remainder Question with Variable Choices

A refresher on remainders follows the solution. 

When integer b is divided by 13, the remainder is 6. Which of the following cannot be an integer?

A) 13b/52
B) b/26
C) b/17
D) b/12
E) b/6



"When b is divided by 13, if the remainder is 6"
Let's take that to mean the following:
b/13 = q remainder 6
q will be an integer since it's essentially the answer to the equation.

Rearrange, remembering that remainders stay the same number when you multiply by the divisor (which is 13 in this case)
b = 13q + 6

Plug in the answers so we can see which works best.

A) 13b/12

Will 13(13q + 6)/12 equal an integer if we plus in for q? 

Let's say q = 6. 

If q = 6, then 13q + 6 = 84, which is is divisible by 12; hence 13(84)/12 is an integer.

B) b/26

Could (13q + 6)/26 result in an integer for an integer value of q? 

First off, 26 is exactly 2 times 13. So, for any integer value of q, 13q will either be divisible by 26 if q is even or produce a remainder of 13 if q is odd. 

Adding 6 to 13q, the expression will either produce a remainder of 6 or 19, but will never produce a remainder of zero. So, (13q + 6)/26 = b/26 can never equal an integer.

B.

GMAT GRE Probability Problem

Set A: {1, 3, 4, 6, 9, 12, 15}

If three numbers are randomly selected from set A without replacement, what is the probability that the sum of the three numbers is divisible by 3?

A. 3/14
B. 2/7
C. 9/14
D. 5/7
E. 11/14

GMAT Inequalities Quant Question

If 4x > y and y< 0 then which of the following could be the value of x/y ?

A)0
B)1/4
C)1/2
D)1
E)2

Hard GMAT Mixture and Ratio Quant Problem

<After the problem solution, there is a refresher on ratios if you would like to review it>


How should a grocer mix 4 types of peanuts, worth 54 cents, 72 cents, $1.2,0, and $1.44 respectively per pound, so that the grocer will make a total mixture which is 96 cents per pound?

(A) 8:4:4:7
(B) 24:12:12:50
(C) 4:8:9:4
(D) 16:42:28:10
(E) Cannot be determined

GMAT Probability - Selection from a group

How many randomly assembled people do u need to have a better than 50% probability that at least 1 of them was born in a leap year?

3
9
27
54
81

700 Level GMAT PS Question

At a sandwich shop, all sandwiches sell for the same price. If the shop were to decrease the price of each sandwich, it would sell 5 more sandwiches per day its daily revenue would remain constant at $150. How much could a sandwich currently cost?

A. $3
B. $4
C. $5
D. $6
E. $10

GMAT, GRE Number Properties I, II, III Must Be True Question

If x and y are distinct prime numbers, each greater than 2, which of the following must be true?

(I) x+y is divisible by 4
(II) x*y has an even number of factors
(III) x+y has an even number of factors

A. I only
B. II only
C. I and III only
D. II and III only
E. I and II only




These kinds of questions usually ask which of the following must be true, or which of the following is always so that it is true no matter what set of numbers you choose to plug in. If you can prove that a statement is not true for one particular set of numbers, it will mean that this statement is not always true and hence not a correct answer. That's the main strategy for solving must be true questions, in a nutshell. 

I. x+y is divisible by 4 

if x=3 and y=7then x+y=10

this is not divisible by 4. So this statement is not always true

II. xy has an even number of factors

only perfect squares have an odd number of factors, 

as xx and yy are distinct prime numbers then xy cannot be a perfect square and thus cannot have an odd number of factors, so xy must have an even number of factors. This statement is always true;

III. x+y has an even number of factors

now, x+yx+y can be a perfect square, for example, if x=3 and y=13 then x+y=16 is a perfect square

so x+y can have an odd number of factors. So this statement is not always true;

Answer: B (II only)

Tips about the perfect square with an example:

1. The number of distinct factors of a perfect square is ALWAYS ODD. The reverse is also true: if a number has the odd number of distinct factors then it's a perfect square;

2. The sum of distinct factors of a perfect square is ALWAYS ODD. The reverse is NOT always true: a number may have the odd sum of its distinct factors and not be a perfect square. For example: 2, 8, 18 or 50;

3. A perfect square ALWAYS has an ODD number of Odd-factors and EVEN number of Even-factors. The reverse is also true: if a number has an ODD number of Odd-factors, and an EVEN number of Even-factors then it's a perfect square. For example, odd factors of 36 are 1, 3, and 9 (3 odd factors) and even factors are 2, 4, 6, 12, 18, and 36 (6 even factors);

4. Perfect square always has even powers of its prime factors. The reverse is also true: if a number has even powers of its prime factors then it's a perfect square. For example, 36=22∗32$36=2^2\ast 3^2$, powers of prime factors 2 and 3 are even.

NEXT:
There is a formula for Finding the Number of Factors of an Integer:

First make prime factorization of an integer n=ap∗bq∗cr$n=a^p\ast b^q\ast c^r$, where a$a$, b$b$, and c$c$ are prime factors of n$n$ and p$p$, q$q$, and r$r$ are their powers.

The number of factors of n$n$ will be expressed by the formula (p+1)(q+1)(r+1)$(p+1)(q+1)(r+1)$. NOTE: this will include 1 and n itself.

Example: Finding the number of all factors of 450: 450=21∗32∗52$450=2^1\ast 3^2\ast 5^2$

The total number of factors of 450 including 1 and 450 itself is (1+1)∗(2+1)∗(2+1)=2∗3∗3=18$(1+1)\ast (2+1)\ast (2+1)=2\ast 3\ast 3=18$ factors.

Back to the original question:

Is the positive integer N a perfect square?

(1) The number of distinct factors of N is even --> let's say n=ap∗bq∗cr$n=a^p\ast b^q\ast c^r$, given that the number of factors of n$n$ is even --> (p+1)(q+1)(r+1)=even$(p+1)(q+1)(r+1)=even$. But as we concluded if n$n$ is a perfect square then powers of its primes p$p$, q$q$, and r$r$ must be even, and in this case number of factors would be (p+1)(q+1)(r+1)=(even+1)(even+1)(even+1)=odd∗odd∗odd=odd≠even$(p+1)(q+1)(r+1)=(even+1)(even+1)(even+1)=odd\ast odd\ast odd=odd\ne even$. Hence n$n$ cannot be a perfect square. Sufficient.

(2) The sum of all distinct factors of N is even --> if n$n$ is a perfect square then (according to 3) sum of odd factors would be odd and the sum of even factors would be even, so the sum of all factors of the perfect square would be odd+even=odd≠even$odd+even=odd\ne even$. Hence n$n$ cannot be a perfect square. Sufficient.

Answer: D