Arithmetics (Number Properties)

Question 1

1) Definition of an even integer
2) Write the expression of consecutive even integers

Answer 1

1) an integer that is divisible by 2,
2) 2n (with n=0,1,2,3,..k)

Question 2

1) Definition of an odd integer
2) Write the expression of consecutive odd integers

Answer 2

1) any integer that is not divisible by 2, written as 2n+1
2) 2n + 1 (with n=0,1,2,3,..k)

Question 3

1) Definition of a prime number
2) What is the general formula for prime number?

Answer 3

1) a positive integer that has exactly two different positive divisors (factors): 1 and itself (2,3,5,7,11,13..)
2) 6n +1 or 6n-1 (except for 2 and 5)

Question 4

What is the smallest prime number?
Is 1 a prime number?

Answer 4

2
no since it only has one divisor

Question 5

Definition of integer

Answer 5

any whole negative or positive number, including 0 (that they are not fractions.), even or odd

Question 6

Is 0 an odd or an even number?

Answer 6

0 is an even number

Question 7

If a is factor b and a is factor of c, then
If a is factor of b and b is factor of c, then

Answer 7

a is factor of (b + c) or (b- c)
a is factor of c

Question 8

even +/- even =?
even +/- odd = ?
odd +/- odd = ?

Answer 8

even
odd
even

Question 9

even * even = ?
even * odd = ?
odd * odd =?

Answer 9

even
even
odd

Question 10

An integer is divisible by 3 if
An integer is divisible by 6 if
An integer is divisible by 9 if

Answer 10

the SUM of the integer’s DIGITS is divisible by 3
the integer is divisible by BOTH 3 and 2
the SUM of the integer’s DIGITS is divisible by 9

Question 11

An integer is divisible by 4 if
An integer is divisible by 8 if
An integer is divisible by 12 if

Answer 11

the integer is divisible by 2 TWICE, or if the LAST TWO digits are divisible by 4
the integer is divisible by 2 THREE TIMES, or if the LAST THREE digits are divisible by 8
the integer is divisible by BOTH 3 and 4

Question 12

An integer is divisible by 5 if
An integer is divisible by 10 if

Answer 12

the integer ends in 0 or 5
the integer ends in 0

Question 13

An integer is divisible by 7 if

Answer 13

you take the LAST digit, DOUBLE it, and SUBTRACT it from the rest of the number, if the answer is divisible by 7 ( including 0) then the number is divisible by 7

Question 14

An integer is divisible by 11 if

Answer 14

you SUM every SECOND digit and then SUBSTRACT all other digits and the answer is 0 or divisible by 11, then the number is divisible by 11

Question 15

How do you express 15 is divided by 6 in math? What other expression can you write?

Answer 15

15/6
6 divides 15

Question 16

Composite number
Is 0 a composite number?
Is 0 positive or negative number?

Answer 16

• Number that has more than two factors. Therefore, composite number is non-prime number
• NO
• It is neither

Question 17

If x and y are positive integers, there exist unique integers q (the quotient) and r (remainder) such that

Answer 17

y = x (divisor) . quotient + remainder
or y/x = quotient + remainder/x

Note: both quotient & remainder are integers

Question 18

How to check whether 318 is divisibly by 6?

Answer 18

1) 318 is even number so divisible by 2
2) 3+1+8= 12 is divisible by 3
Hence, 318 is divisible by 6

Question 19

T or F. If you have a same divisor for two integers, you can not add and subtract their remainders directly

Answer 19

False. For example:
If x leaves a remainder of 4 after division by 7 and z leaves a remainder of 5 after division by 7. Algebraically,
x = 7q1 + 4
z = 7q2 + 5
- x + z = 7 (q1+q2) + 9 = 7 (q1+q2) + 7 +2 so x+ z is multiple of 7, plus 2 (remainder)
- x - z = 7 (q1 - q2) + 4 -5 = 7 (q1 - q2 ) - 1 = 7 (q1 - q2) - 7+6 so x-z is multiple of 7, plus 6 (remainder)

Question 20

T or F. If you have a divisor throughout, you can multiply remainders directly, as long as you correct excess or negative remainders

Answer 20

T - for i.e:
- When x, z divide 7, leaving the 4 & 5 as remainder, respectively, x.z will have the remainder 20 .
- Taking out excess of 7 twice, we have 6 left. Thus, remainder of x.z is 6

Question 21

T or F: When we classify a number odd or even, we mean that number can be either integer or non-integer

Answer 21

F.
Odd or even number must refer to integers only

Question 22

What does it mean when we say 3 goes into 12 evenly

Answer 22

3 is a divisor/factor of 12
12 is divisible by 3
3 must divide 12

Question 23

How do you express “ m is a multiple of n”
T or F. An integer can have more factors than its multiple?

Answer 23

m = k.n. (i.e: 12 is a multiple of 3)
F - 8 only has four factors while multiple of 8 is unlimited

Question 24

If you add/subtract a multiple of N to a non-multiple of N, the result is
If you add/subtract two non-multiple of N, the result is

Answer 24

• non- multiple of N
• could be either a multiple of N or a non-multiple of N

Question 25

How do you find the greatest common factor (divisor)?

Answer 25

1) Prime- factorization 2) Of all integers, Multiply the common factors that have the lowest power Ex: GCF of 120 and 100 is 20

Question 26

if the sum of the digits of x is equal to 21, you can infer that x is divisible by?

Answer 26

by 3 but not 9

Question 27

In combinatorics, **w/o replacement**, there is how many to arrange n distinct objects? Why?

Answer 27

n! - for i.e: 3 letters A,B & C have 3x2x1 = 3! - 3 options for 1st spots - 2 options for 2nd - 1 option for last slot

Question 28

If only k objects (w/o further arranging among them) are to be selected from a larger set of n objects, then the number of combination for k objects is ____

Answer 28

Total n! / (SELECTED! STAYED!) or : nPk = n! / {k! (n - k)!}

Question 29

How do you find the lowest (least) common multiple- LCM? Where do you often need to do calculation of LCM?

Answer 29

1) Prime- factorization 2) Multiply all prime factors but only choose the highest power of common factors Ex: GCF of 120 and 100 is 600 3) find the common multiple of two fractions' denominator

Question 30

What is unconventional way to find LCM (a,b) ?

Answer 30

(a x b)/ GCF (a,b)

Question 31

Is 132,300 a perfect square?

Answer 31

No because: 132,300 = 2^2.3^3. 5^2. 7^2 - Prime factorization: extra 3 while other prime numbers is square

Question 32

7,654.321 List the position of each digit in English term

Answer 32

7 = thousands 6= hundreds 5 = tens 4 = ones of units 3 = tenths 2= hundredths 1= thousandths

Question 33

What is the shortcut for decimal places in fraction number with exponent? i.e: (0.2) ^ 6 = ?

Answer 33

The number of decimal places of new number = the value of the exponent x the number of decimal places in the original decimal 1x6= 6 decs —move from the last digit (after exponent) to the dot 6 numbers — 0.000032

Question 34

Can you apply decimal places short cut to convert 0.00005 to the value before its squared/cubed?

Answer 34

No because the # of decimal places is not divisible by 2 or 3 (exponent values) or other integer exponent value

Question 35

# In combinatorics, "or" mean "and" mean

Answer 35

"add" "multiply"

Question 36

If we pick k members from one group how many arrangements are there for k members (w/ order for k members) ?

Answer 36

n! / (n-k)!

Question 37

What are the prime numbers in the denominator to have a fraction as a terminate decimal?

Answer 37

the prime factorization must consist of only 2's or only 5 or only 2 and 5 Ex: 3/105 doesn't have terminate decimal

Question 38

How to calculate the total prime factors (length)?

Answer 38

Multiple the addition of 1 to the value of exponents of all prime factors - i.e: If M has prime factorization a^x . b^y . c^z , the total number of different factors of M is (x+1). (y+1). (z+1)

Question 39

T or F. The remainder must be always smaller than the divisor

Answer 39

T

Question 40

What are the possible remainders when an integer divide by a positive integer N? (p.100 Manhattan- Number Properties)

Answer 40

0 -> N - 1 i.e: If N =4, there are 4 remainders (0,1,2,3)

Question 41

In combinatorics, without replacement implying....

Answer 41

you can't put the objects back into the pool - for i.e: 3 letters A,B & C will have 3! instead of 3x3x3 = 27 (w/ replacement)

Question 42

The cool trick: Squaring numbers with unit digits 5 (i.e: 25^2, 65^2, 105^2)

Answer 42

1) Multiply 1st number with the next number of one unit greater 2) Add 25 to the end i.e: 25^2 = 625 (2x3) 65^2 = 4225 (6x7) 105^2 =11025 (10x11)

Question 43

what is least common multiple (LCM)? What is the greatest common factor (GCF)?

Answer 43

The lowest multiple that two numbers have in common (the denominator in fraction) i.e: 4 & 6 has 12 as the LCM The highest value of a factor that two numbers have in common i.e: 4 & 6 has 2 as the GCF

Question 44

If **an object** is to be chosen from a set of **m objects** and a second object is to be chosen from a different set of **n objects**, how many ways of choosing both objects simultaneously?

Answer 44

m x n ways

Question 45

What is the remainder of an integer divided by 10?

Answer 45

The remainder will be the same as the units digit of the original number

Question 46

T or F: "B goes into A" means that all of the prime factors of B are also prime factors of A

Answer 46

True In other words, A is divisible by B I.e: 3 goes into 12

Question 47

What is trailing zeros?

Answer 47

The number of zeros that follows the non-zero digits

Question 48

# What is the implication of this statement? which of the following must be true?

Answer 48

It means that either: 1) There is only one correct answer 2) The value of the correct answer must always satisfy the given condition

Question 49

if the least common denominator of x/y and 1/3 is 6, what is the value of y?

Answer 49

y has two values: 2 or 6 (x/6 & 1/3 vs x/2 & 1/3)

Question 50

Is x^2 greater than x ? (1) x^2 is greater than 1. (2) x is greater than –1.

Answer 50

1) sufficient via test cases 2) X> -1 but X can fall into between 0 & 1, in which x^2 < x (not sufficient)

Question 51

# What is unique about this question? Can the positive integer n be written as the sum of two different positive prime numbers? (1) n is greater than 3. (2) n is odd.

Answer 51

The right answer must be the case that either always "YES" or always "NO - not both 1) n > 3: n=4 can't be written ("NO") as such while n =5 can be ("YES") -> NOT sufficient 2) same thinking as (1) n = 11("NO") can't be written vs n = 5 ("YES") -> NOT sufficient

Question 52

# What is the broad concept applicable in other scenarios? What is the thousandths digit in the decimal equivalent of 53/5000?

Answer 52

when denominator is 2 or 5: 1) Terminal decimals 2) Easy conversion to percent

Question 53

How to quickly find the repeating pattern of the decimal form of 6/11?

Answer 53

1) Find a way for the denominator to appear 99 (i.e: 6/11= 54/99) 2) Whatever value in the numerator is the repeating digits. (0.545454)

Question 54

# What are different ways to solve? In a certain class consisting of 36 students, some boys and some girls, exactly 1/3 of the boys and exactly 1/4 of the girls walk to school. What is the greatest possible number of students in this class who walk to school? A.9 B.10 C.11 D.12 E.13

Answer 54

1) We can do max/min range to eliminate A,D & E - Either ALL boys or ALL girls-> 9 girls or 12 boys walk; however this can't be our answer because the stimulus includes some B&G - Backsolve choice B & C w/ aglebra 2) We can set up algebra: Walk = 1/3 B + (36-B) 1/4 = B/12 + 9. Since B is multiple of 12, to maximize walk, we maximize B = 24

Question 55

# Beside doing test cases, set up algebra function to solve The remainder is 3 when x + 1 is divided by 4. What is remainder when x is divided by 4?

Answer 55

x + 1 = 4q + 3 -> x = 4q +2 so, the remainder will be 2 everytime x is divided by 4

Question 56

# Use LOGICS w/o test cases y = 3x + 2. Is y divisible by 6?

Answer 56

It means that y when divided by 3 gives us a quotient and 2 as remainder. So y is not completely divisible by 3, and hence can't be divisible by 6.

Question 57

# You can easily be confused with double matrix, find out why In a nationwide poll, N people were interviewed. If 1/4 of them answered "yes" to question 1, and of those, 1/3 answered "yes" to question 2, which of the following expressions represents the number of people interviewed who did NOT answer "yes" to both questions?

Answer 57

Common sense: 1-1/4 + 1/4 x 2/3 = 3/4 + 1/6 = 11/12

Question 58

# What kind of reasoning that this problem teach us? The weights of four packages are 1, 3, 5, and 7 pounds, respectively. Which of the following CANNOT be the total weight, in pounds, of any combination of the packages? a) 9 b) 10 c)12 d) 13 e) 14

Answer 58

The sum of all members - The sum of a subgroup = Value of Individual - i.e: Σ = 16 & 16 - 12 = 6 = Σ of 1 & 5 lbs subgroup - Hence, Σ = 16 & 16 - 14 (combination of some members) = 2 but there isn't value of 2 in the group so 14 can't be the value of any combination

Question 59

# Rephrase the question If n is a positive integer and n^2 is divisible by 72, then the largest positive integer that must divide n is

Answer 59

1) Rephrase “largest integer that must divide n" -> the greatest factor ALWAYS the case for n 2) patterns of value of n =12k (w/ k =1,2,3..) = 12, 24, 48.. - anything greater than 12 can’t be the answer ( for i.e: if x = 24 & n=12 we know 24 can’t be the factor)

Question 60

# Use LOGICS w/o test cases If the remainder when a certain integer x is divided by 5 is 2, then each of the following could also be an integer, EXCEPT a) x/17 b) x/11 c) x/10 d) x/6 e) x/3

Answer 60

X/10 stand out as the possible answer. Assuming if X/10 has an integer, then X must be divisible by 5 and hence contradict w/ the given stimulus (remainder is 2 when x is divided by 5). Therefore, x/10 can't be an integer.

Question 61

what is the characteristic of divisbility for three consecutive integers?

Answer 61

their multiplication is always divisible by 6

Question 62

# Key Lesson If m and n are both two digit numbers and m - n = 11x, is x an integer? (1) The tens digit and the units digit of m are the same (2) m + n is a multiple of 11

Answer 62

1) Re-phrase backward from the fact: m - n = Integer = 11x - X is an integer when m - n is a multiple of 11 2) Test cases for each statement, then in combination - m must be a multiple of 11 & since m + n is multiple of 11, n is also must be

Question 63

# What is the key lesson for this problem? November 16, 2001, was a Friday. If each of the years 2004, 2008, and 2012 had 366 days, and the remaining years from 2001 through 2014 had 365 days, what day of the week was November 16, 2014 ?

Answer 63

Logic: Nov 16 2001 -> Nov 16 2002 = 365 days (with the pattern of 1 week/Fri) = 52 weeks + 1 extra day - Hence: 1 extra day = Saturday which falls on Nov 16 2002 -> Friday will be Nov 15 2002 Apply the divisbility rule, we can draw the pattern: - 10 non-leap years * 1 day forward each = 10 days forward - 3 leap years * 2 days forward each = 6 days forward -> 10 + 6 = 16 days forward = 2 weeks and 2 days extra 2 weeks forward takes us to Friday, which is added 2 extra days = Sunday on Nov 16 2014

Question 64

# Use LOGICs w/o test cases When the integer n is divided by 6, the remainder is 3, Which of the following is NOT a multiple of 6? (A) n – 3 (B) n + 3 (C) 2n (D) 3n (E) 4n

Answer 64

- If you remove 3 from n, there will be only groups of 6 leftover. So (n - 3) will be divisible by 6. - If you add 3 to n, the remaining will also form a complete group of 6 and hence there will be only groups of 6. So (n + 3) will be divisible by 6. - When you double n, 3 left over will become 6 and so we will have only groups of 6. Hence 2n will be divisible by 6. - If 2n is divisible by 6, 4n must be divisible by 6 too.

Question 65

# how many number of non-zero digits required to satisfy: fewer than 8 zeroes between the decimal point and the first nonzero digit in the decimal expansion of (t/1000)^4

Answer 65

The number of zeroes in between = Total number of decimals (the exponent value) - The number of digits after exponential value of the integer - Since (t/1000)^4 = t^4/ 10^12 and we need 7 or fewer zeroes then 12-7 = at least 5 digits integer

Question 66

# What is this problem testing? If it is 6:27 in the evening on a certain day, what time in the morning was it exactly 2,880,717 minutes earlier? (Assume standard time in one location)

Answer 66

Logics: 6:2**7** minus 2,880,71**7** in any way must end with 0 or in backward thinking: - The rewind time must end in 0, so that added minutes can end in 7

Question 67

# What is the strategy in this problem? How many integers between 324,700 and 458,600 have tens digit 1 and units digit 3?

Answer 67

1) Recognize the pattern by listing out few scenarios that fit the requirement tens digit 1 & units digit 3: - 324,713 ; 324,813 ; 324, 913......458,513 2) Apply the sequence formula for incremental of 100 units above - 458,513 = 324,713 + 100 (n-1) -> n = 4586 - 3247 +1 = 1,339

Question 68

# T or F: In a sequence or exponents, to find the units digit of the product or a sum of integers, pay attention to the pattern that occurs in sequential terms

Answer 68

True

Question 69

What does it mean "an integer in base 10 notation"?

Answer 69

Based 10 notation, or decimal notation, is just a way of writing a number using 10 digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 (usual way), in contrast, for example, to binary numeral system (0 or 1 notation.)

Question 70

# What is the overall strategy for similar pattern problem? If (10^50) – 74 is written as an integer in base 10 notation, what is the sum of the digits in that integer?

Answer 70

1) Draw out cases (analogies) for smaller number to see any pattern 2) Apply that general pattern to the main question