GRE Math Chapter 10-Arithmetic and Number Properties
An integer is divisible by 4 if its digits --
If its last two digits are a multiple of 4 48 4 is a multiple of 4 4 and 1=4 4 and 2=8
Multiply 0.029 by 10^3
Move the decimal point the same number of places to the right as the value of the exponent 29
How would you solve this? (3^2)^4
Multiply 4 and 2 3^8
Is 1 considered a prime number?
No because it cannot be divided by the rules applied to prime factors. 1 and itself
When integer a is divided by 5, the remainder is 2. When integer b is divided by 5, the remainder is 3. What is the remainder when a X b is divided by 5?
Numbers that leave a remainder of 2 when divided by 5 are numbers that are 2 greater than a multiple of 5. Likewise with 3. Ie: 5+2=7 5+3=8 7/5=remainder of 2 8/5= remainder of 3 7x8=56 56/5 11x5=55 remainder of 1 1 is the answer
What do integers include?
Positive and negative whole numbers and zero
An integer that is greater than 1 that has only two factors: 1 and itself
Prime number ie: 2 1 and 2
√xy is a prime number, xy is even, and x>4y>0 Quantity A: y Quantity B: 1 C D
Quantity B ?? page 542
ab>0 bc<0 Quantity A: ac Quantity B: 0
Quantity B A and B have to be the same signs A and B could be both positive A and B could be both negative B and C have to be different signs to be negative. If B is positive, C will be negative. If B is negative, C will be positive. A is basically the same a B and two different signs= a negative number, so quantity B is greater.
a, b and c are positive even integers such that 8>a>b>c Quantity A: the range of a,b and c Quantity B: the average (arithmetic mean) of a, b and c
Since they are positive even integers, this problem can be solved. The answer is C 6-2=4 12/3=4
How do you find the least common multiple of 6 and 8?
Step 1: Determine the prime factorization of each integer {2,3,5,7,9,13,17,19,23,29} 6= *2* and 3 8= *2* and 2 and 2 Step 2: Write out each prime number the maximum number of times that it appears in any one of the prime factorizations ^^^ Step 3: Multiply those prime numbers together to get a least common multiple of the original integers 2,3,2,2 = 24
What are factors?
The divisors of a number. For example, The factors of 36 are: (1)(36) (18)(2) (3)(12) (4)(9) (6)(6)
How many two digit multiples of 6 are multiples of 15?
The least common multiple of 6 and 15 is 30. Every multiple of 30 will also be a multiple of both 6 and 15. That means there are only three two-digit multiples of both 6 and 15 30, 60 and 90 that is because the next one is 120 and it asks for two-digit multiples. 120 is three digits
How do you multiply and divide radical numbers?
To multiply or divide one radical by another, multiply or divide the numbers outside the radical signs, then the numbers inside the radical signs Ie: (6 √3) 2√5= 6 x 2=12 √3 x √5=15 12 √15
How would you solve a problem that has different bases but the same power? Ie: (3^2) (5^2)
To multiply two powers with different bases but the same power, multiply the bases together and raise to the power (3 x 3) (5 x 5) (3x5) (3x5) = 15^2
What is significant about raising 10 to a power?
When 10 is raised to an exponent that is a positive integer, that exponent tells how many zeros the number would contain if it were written out. Ie: 10^6 1,000,000
What would be the rules to solve this? 2√3 + 4√2 - √2 - 3√3
When adding and subtracting radicals, you can only do it to like radicals so 2√3 + 4√2 - √2 - 3√3 (4√2 - √2) + ( 2√3- 3√3) 3√2 + -1√3 3√2 + -√3 OR 3√2 -√3
Divide 416.03 by 10,000
When you divide, you move to the left Since there are 4 zeros, you move 4 to the left .041603
How would you know how to solve this: Multiply: 0.029 by 1,000
You count the number of zeros and then that is how much you move the decimal Ie 0.029 by 1,000 3 zeros 29
How would you solve a problem with a negative exponent? Ie: 5^-3
You would make it into a fraction 1/5^3 1/125
An integer is divisible by 3 if its digits ----
add up to a multiple of 3 ie: 12 1+2=3 12/3=4
What is the largest prime factor of 46,000?
any 0 can be broken down into 5X2 Since there are three 0's (5X2) (5X2) (5X2) Now you just have to break down 46 prime numbers: 2,3,5,7,11,13,17,19,23,29 46 can be broken down into 23 and 2 The largest factor of all of the answers is then 23
What does inclusive mean in math?
containing as part of a whole
An integer is divisible by 2 if its last digit is ---
divisible by 2 ie: 44 44/2 22
One of the members of a set is called an ---
element
raising an even number to any exponent gives a even or odd number
gives an even number Ie: 8^5 =32,768
raising an odd number to any exponent gives a even or odd number
gives an odd number Ie: 5^8 =390,625
When is an integer divisible by 9?
if its digits add up to a multiple of 9 63 6+3=9 9x7=63
The product of a specified number and an integer is called a
multiple For example, {3, 12, 90} are all multiples of 3. 3x1= *3* 4x3=*12* 30x3=*90* So, the multiples is the end product. Multiples of 3 means that the end product came from something being multiplied by 3.
This number is neither negative nor positive
zero
What would be the best way to solve: (1/2)^2
(1/2) (1/2) 1/4
How would you do the prime factorization of 210?
(2) (105) 105= 3 (35) (2) (3) 35= 5 and 7 (2)(3)(5)(7)
What is the value of 4³ x 9³/6⁵
(4x9)³/ 6⁵ 36³ / 6⁵ (6²)³=6⁶ / 6⁵ 6⁶-6⁵=6 Answer is 6
An integer is divisible by 5 if its last digit is
0 or 5 55/5=11 60/5=12
5^0
1 Raising a number to the zero power is always 1
If x is a positive integer, which one of the following could be the remainder when 73^x is divided by 10? 0 1 2 3 4 5 6 7 8 9
1,3,7,9 As with multiplication, when an integer is raised to a power, the units digit is determined solely by the product of the units digits. Those products will form a repeating pattern. Here, 3¹=3 3²=9 3³=27 3⁴=81 3⁵=24*3* Here the pattern returns to its original value of 3 and any larger power of 3 will follow this same pattern: 3,9,7 and then 1.
What is the scientific notation of 1.75 million?
1.75 X 10⁶
If m is the product of all the integers from 2 to 11, inclusive, and n is the product of all the integers form 4 to 11, inclusive, what is the value of n/m
1/6 Here, you know that you can cancel bc m=2,3,4,5,6,7,8,9,10,11 n=4,5,6,7,8,9,10,11 everything cancels besides 2 and 3 that you can then multiply
What is the greatest common factor of 36 and 48?
12
On a trip, Joe's car traveled an average of 36 miles per gallon of fuel. Approximately how many kilometers did the car travel on 10 liters of fuel? (5 miles= approximately 8 kilometers; 1 gallon= approximately 4 liters)
144 kilometers 36 miles/1 gallon X 8 kilometers/ 5 miles = 288 kilometers/5 gallons 288 kilometers/5 gallons X 1 gallon/4 liters = 288 kilometers/20 liters 14.4 kilometers/1 liter The car has 10 liters of fuel in the tank. 10 liters X 14.4 =144 kilometers
Molly worked at an amusement park over the summer. For two weeks, she was paid according to the following schedule: at the end of the first two weeks, she received $160. At the end of each subsequent 2-week period, she received $1 plus an additional amount equal to the sum of all payments she had received in previous weeks. How much money was Molly paid during the full 10 weeks of summer?
1st pay period: 160 2nd pay period: 160+161=321 3rd pay period: 321+ 322=643 4th pay period: 643 + 644= 1287 5th pay period: 1287 + 1288= 2575
The smallest prime number
2
What are the distinct numbers of the set (2, 2, and 3)
2 and 3 are distinct numbers because they are different. 2 and 2 are not distinct numbers because they are not different
What are the first 10 prime numbers?
2,3,5,7,11,13,17,19,23,29
If √x is an integer and xy²=36, how many values are possible for the integer y? A) Two B) Three C) Four D) Six E) Eight
26) E Go to page 541
If p is the sum of all the integers from 1 to 150, inclusive, and q is the sum of all the integers from 1 to 148, inclusive, what is the value of p-q?
299 The median of 150= 75.5 75.5 (150)=11,325 The median of 148= 74.5 74.5 (148)= 11,026 11,325-11,026= 299
Rules of exponents and operations: (2^2) (2^3) What is the end exponent?
2^5 When multiplying exponents, you add them if they have the same base 2+3=5
Name these placeholder numbers 315. 246
315.246 3= hundreds 1= tenths 5= ones 2=tenths 4=hundreds 6= thousands
The average (arithmetic mean) of 11 integers is 35. What is the sum of all the integers?
385 the 11 integers/11=35 35 (11)= 385
What is the scientific notation of 4.5 billion?
4.5 X 10⁹
Rules of exponents and operations: 4^5 divided by 4^2 What is the end exponent?
4^3 5-2=3
What is the remainder of 17?
5 and 3 = 15 the remainder is 2
If set S consists of all positive integers that are multiples of both 2 and 7, how many numbers in set S are between 140 and 240, inclusive
8 A positive integer that is a multiple of both 2 and 7 is a multiple of 14. Since 140 is a multiple of 14, start listing there and count the terms in the range {140, 154, 168, 182, 196, 210, 224, 238}
The product of positive integers f, g, and h is even and the product of integers f and g is odd Quantity A: The remainder when f is divided by 2 Quantity B: The remainder when h is divided by 2 C D
A even, even, even= even odd,odd,odd=odd even,even,odd=even even, odd,odd=even odd,odd=odd Therefore, we know that f and g have to be odd and then we can determine that the pattern is even, odd, odd---- SO, h is even. When an even number is divided by 2, there will be NO remainder When an odd number is divided by 2, there will be a remainder. So, the answer is A
What two things must be multiplied to make a multiple?
A multiple is made up of an integer and a specific number
When is an integer divisible by 6?
An integer is divisible by 6 if it is divisible by both 2 and 3
Quantity A: The remainder when 10¹¹ is divided by 2 Quantity B: The remainder when 3¹³ is divided by 3 C D
C Because 10 is a even number, so it 10¹¹ and 0 is the remainder. Similarly, 3¹³ is a multiple of 3, and 0 is the remainder. Therefore, the quantities are equal
a,b and c are integers such that a<b<c quantity A: a+b+c/3 quantity B: b C D
D Realize that it does not say that these are consecutive integers. So the relationship cannot be determined
On a certain train, tickets cost $6 each for children and $9 each for adults. The total train ticket cost for a certain group of six passengers was between $44 and $50. Which is greater? Quantity A: The number of children in the group Quantity B: The number of adults in the group C: Equivalent D: Relationship cannot be determined
D There are several possibilities
x>0>y Quantity A: x-y Quantity B: (x+y)² C D
D x could be any number and y could be any number. The relationship cannot be determined
What is the sum of all the integers from 1 to 80 inclusive? A) 3200 B) 3210 C) 3230 D) 3240 E) 3450
D) Multiply the median by the number of terms in the set. The median of the numbers is 40.5 so, 40.5 (80) =3240
If x, y and z are integers, y+z=13 and xz=9, which of the following must be true? A) x is even B) x=3 C) y is odd D) y>3 E) z<x
D) y>3 If xz=9 and z and z must be both integers, then they are 1 and 9 (or -1 and -9) or 3 and 3 (or -3 and -3). Therefore, they are both odd. More generally, the product of two integers will only be odd if the component integers themselves are both odd. Because z is odd, and y+z equals 13 (an odd), y must be even A) x is not even B) x could be 3 but doesn't have to be. C) y is not odd E) z does not have to be less than x
On a number line, the distance from A to B is 4 and the distance from B to C is 5. Quantity A: distance from A to C Quantity B: 9 C D
D: cannot determine Whenever a question looks this straightforward (4+5=9), so the quantites initially appear equal, be suspicious. Draw the number line. If the points A, B and C are in alphabetical order from left to right, then the distance from A to C will be 9. However, alphabetical order is not required.
Which quantity is greater? 5³-5² OR 5
Do not make the mistake of subtracting the exponents. First find 5³=125 5²=25 125-25=100 quantity A is greater than quantity B (100 is greater than 5)
If 0<a<1/b<1 then which of the following must be true? A) a²>a>b>b² B) b>a>a²>b² C) b²>a>a²>b D) b²>a²>b>a E) b²>b>a>a²
E -notice that when you square decimals they become smaller. BUT, b is not a decimal unless it is 1/b. The 1/b makes the decimal.
If x is a non-zero integer and 0<y<1, which of the following must be greater than 1? A) x B) x/y C) xy² D) x²y E) x²/y
E Here, pay attention to the positive and negative values in order to solve for it. A) cannot be an answer because X could be negative and we need greater than positive 1 B) x/y cannot be an answer because again, X could be negative and we need greater than positive 1 C) again, negative x D) could work but we know that y is a decimal. When you multiply a decimal by a whole number, it makes it smaller. Ie: 1 (.5)=.5 E) This works because when you divide by a decimal it creates a larger number. Ie: 1/.5=2
What does distinct factors represent?
Factors that are not the same number
If X is an integer, how many values of x are there such that the absolute value of x is less than 6 AND the absolute value of x is greater than 3.
Here there are 4 values because even though 4 and 5 seem to be the same as -4 and -5. In absolute value, they are not the exact same. Instead, they lead to 4 values.