Arithmetics (Number Properties) Flashcards

1
Q

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

A

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

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2
Q

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

A

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

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3
Q

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

A

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)

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4
Q

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

A

2
no since it only has one divisor

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5
Q

Definition of integer

A

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

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6
Q

Is 0 an odd or an even number?

A

0 is an even number

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7
Q

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

A

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

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8
Q

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

A

even
odd
even

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9
Q

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

A

even
even
odd

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10
Q

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

A

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

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11
Q

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

A

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

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12
Q

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

A

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

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13
Q

An integer is divisible by 7 if

A

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

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14
Q

An integer is divisible by 11 if

A

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

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15
Q

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

A

15/6
6 divides 15

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16
Q

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

A

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

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17
Q

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

A

y = x (divisor) . quotient + remainder

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18
Q

How to check whether 318 is divisibly by 6?

A

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

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19
Q

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

A

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)

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20
Q

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

A

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

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21
Q

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

A

F.
Odd or even number must refer to integers only

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22
Q

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

A

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

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23
Q

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

A

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

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24
Q

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

A

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

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25
Q

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

A

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

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26
Q

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

A

by 3 but not 9

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27
Q

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

A

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

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28
Q

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 ____

A

Total n! / (PICKED! OUT!) or :
nPk = n! / {k! (n - k)!}

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29
Q

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

A

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

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30
Q

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

A

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

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31
Q

Is 132,300 a perfect square?

A

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

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32
Q

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

A

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

33
Q

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

A

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

34
Q

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

A

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

35
Q

what are the results of:

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

110% of the original
25% less than

36
Q

How do you convert 17/25 to percentage?

A

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

37
Q

In combinatorics, “or” mean

“and” mean

A

“add”

“multiply”

38
Q

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

A

n! / k! (n-k)!

39
Q

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

A

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

40
Q

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

A

5/6
5/4
4/3

41
Q

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

A

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

42
Q

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

A

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

43
Q

How to calculate the total prime factors (length)?

A

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 number of different factors of M is (x+1). (y+1). (z+1)

44
Q

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

A

T

45
Q

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

A

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

46
Q

In combinatorics, without replacement implying….

A

you can’t put the objects back into the pool

47
Q

The cool trick: Divide by 5

A

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

48
Q

The cool trick: Multiplying with the decimals ending in 5

A

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

49
Q

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

A

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)

50
Q

What critical thinking are you using?

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

A

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

51
Q

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

A

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

52
Q

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?

A

m x n ways

53
Q

What is the remainder of an integer divided by 10?

A

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

54
Q

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

A

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

55
Q

What is trailing zeros?

A

The number of zeros that follows the non-zero digits

56
Q

What is the implication of this statement?

which of the following must be true?

A

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

57
Q

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

A

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

58
Q

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

A

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

59
Q

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.

A

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

60
Q

What is the broad concept applicable in other scenarios?

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

A

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

61
Q

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.

A

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

62
Q

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

A

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)

63
Q

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

A

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

64
Q

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?

A

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

65
Q

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

A

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.

66
Q

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?

A

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

67
Q

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

A

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)

68
Q

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

A

1) Rephrase “largest integer that must divide n” -> find the value 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 since it can’t divide the minimum value of n=12 (or the largest value divide 12 is 12)

69
Q

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

A

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.

70
Q

what is the characteristic of divisbility for three consecutive integers?

A

their multiplication is always divisible by 6

71
Q

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

A

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

72
Q

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.

A

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

73
Q

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 ?

A

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

74
Q

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

A
  • 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.
75
Q

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

A

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

76
Q

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)

A

Logics: 6:27 minus 2,880,717 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

77
Q

What is the strategy in this problem?

How many integers between 324,700 and 458,600 have tens digit 1 and units digit 3?

A

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

78
Q

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

A
79
Q

What is the key lesson?

When 24 is divided by the positive integer n, the remainder is 4. Which of the following statements about n must be true?
(I) n is even (II) n is a multiple of 5. (Ill) n is a factor of 20.

A

Concept:
- the remainder is always less than the divisor, so n > 4: n can’t be 2 or 4
- Since n can be 5 & statement is about must be true: (I) can be eliminated