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 even 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

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) - 1 = 7 (q1 - q2 - 1) - 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:
1) When x, z divide 7, leaving the 4 & 5 as remainder, respectively, x.z will have the remainder 20 .
2) 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 n

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, with replacement, there is how many to arrange n distinct objects?

Answer 27

n!
i.e: 3 letters A,B & C have 3! = 6

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! / (PICKED! OUT!) 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

what are the results of:

• 10% greater than original
• 75% of the original

Answer 35

110% of the original
25% less than

Question 36

How do you convert 17/25 to percentage?

Answer 36

Multiply by 4 for numerator and denominator. We have 68/100 = 68%

Question 37

In combinatorics, “or” mean

“and” mean

Answer 37

“add”

“multiply”

Question 38

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

Answer 38

n! / k! (n-k)!

Question 39

11.1 % equal to fraction of
12.5% equal to fraction of
16.7% equal to fraction of
175% equal to fraction of

Answer 39

1/9
1/8
1/6
7/4

Question 40

83.3% equal to fraction of
125% equal to fraction of
133% equal to fraction of

Answer 40

5/6
5/4
4/3

Question 41

T or F. You can only pick one smart number for one variable.

Answer 41

F - Depend on the lvl difficulty problems, sometimes you must know where to pick 2 or 3 smart numbers

Question 42

What does prime factorization in the fraction denominator (in fully reduced form) must consist in order to have a fraction as a terminate decimal?

Answer 42

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 43

How to calculate the total prime factors (length)?

Answer 43

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

Question 44

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

Answer 44

T

Question 45

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

Answer 45

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

Question 46

In combinatorics, without replacement implying….

Answer 46

you can’t put the objects back into the pool

Question 47

The cool trick: Divide by 5

Answer 47

1) Double the dividend
2) Move decimal one place to the left
i.e: 8/5 = 1.6
435/5 = 87.0

Question 48

The cool trick: Multiplying with the decimals ending in 5

Answer 48

Multiplying by halving and doubling:
1) 16x 5.5 = 8 x11 = 88
2) 2.25x 36 =4.5x18 = 9x9 = 81

Question 49

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

Answer 49

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 50

The “prime sum” of an integer n greater than 1 is the sum of all the prime factors of n, including repetitions . For example, the prime sum of 12 is 7, since
12= 2x2x3and2+2+3= 7.For which of the following integers is the prime sum greater than 35 ?

(A) 440 (B) 512 (C) 620 (D) 700 (E) 750

What critical thinking are you using?

Answer 50

1) Start with easiest prime factorization to eliminate A,D, E
2) Between B & C, start with C first.

Question 51

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

Answer 51

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 52

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 52

m x n ways

Question 53

What is the remainder of an integer divided by 10?

Answer 53

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

Question 54

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

Answer 54

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

Question 55

What is trailing zeros?

Answer 55

The number of zeros that follows the non-zero digits

Question 56

What is the implication of this statement?

which of the following must be true?

Answer 56

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 57

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

Answer 57

y has two values: 2 or 3

Question 58

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

Answer 58

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

Question 59

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 59

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 60

What is the broad concept applicable in other scenarios?

What is the thousandths digit in the decimal equivalent of 53/5000?

Answer 60

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

Question 61

Why you need to be careful about this problem?

If A = 2B, is A^4 > B^4?
(1) A^2 = 4B^2.
(2) 2A + B < A/2 + B.

Answer 61

There is no restriction of A &B, hence they can be any types of value: non-integers, pos/neg or equal to 0

Question 62

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

Answer 62

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 63

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 63

1) We can do max/min range to eliminate A,D & E (i.e: B or G =0 -> 9 or 12 students walk; however this can’t be our answer because the stimulus requires both B & G) -> Backsolve choice B & C
2) We can set up algebra: Walk = 1/3 B + (30-B) 1/4 = B/12 + 9. Since B is multiple of 12, to maximize walk, we maximize B = 24

Question 64

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 64

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

Question 65

y = 3x + 2. Is y divisible by 6? Use logics W/o using test cases

Answer 65

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 66

practice w/o double matrix

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 66

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

Question 67

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 67

The sum of all members - The sum of subgroup = Value of Individual Member/ The sum of other subgroup (i.e: Σ = 16 & 16- 14 = 2 but there isn’t value of 2 in the group so 14 can’t be the value of any combination)

Question 68

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 68

1) The phrase “integer that must divide n” really means “n must be divisible by that integer.” -> find patterns of value of n
2) Using prime factorization we know that n =12k (w k =1,2,3..) = 12, 24, 48.. n is min at 12 but isn’t divisible by 24 or 48 so 12 must be the largest integer divisible by n (GCF)

Question 69

W/o using test cases, can you find the answer by the outlook of answers?

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 69

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 70

what is the characteristic of divisbility for three consecutive integers?

Answer 70

their multiplication is always divisible by 6

Question 71

What did you assume wrong to claim that X must always be integer

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 71

1) First reaction: The difference must be an integer, and if that integer = 11x then we can assume x must be an integer. However, since there are such cases that x= 30/11 & the difference is still an integer, my initial assumption is wrong. I’m supposed to prove that x is an integer.
2) Rephrase question: When the question is asking us whether x an integer, it means to ask if m-n is a multiple of 11

Question 72

What are the key lessons for this problem?

If j and k are integers and jk=12, what is the value of k?
(1) j/6 is an integer.
(2) k/2 is an integer.

Answer 72

1) in DS, very be wary when the question ask about the specific value for multiplication/division - Always do the exhaustive test cases: negative/non-integer/zero….
2) In this problem, k can exist as pos/neg values, and hence we can’t determine one solution of k

Question 73

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 73

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 74

W/o test case, use logic to solve

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 74

• 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.