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GRE > Integer Properties > Flashcards

Integer Properties Flashcards

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1
Q

Factor

A
  • a number multiplied to get an integer
  • In A * B = C, A and B are factors of C
  • 1 is a factor of every integer: 1 * 5 = 5
  • Every integer is a factor of itself: 7 * 1 = 7
  • Every integer has at least 2 factors
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2
Q

Divisor

A
  • an integer that divides another integer to get an integer quotient
  • In C/A = B, A is a divisor of C
  • There is no difference between a factor and a divisor
    • 8 is a factor of 24 and a divisor of 24
  • Similarly
    • 8 is not a factor or divisor of 12, because it results in a fraction, 12/8
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3
Q

Finding Factors (Simple)

A
  • List the factor pairs
    • Factors of 36
      • 1 & 36, 2 & 18, 3 & 12, 4 & 9, 6 & 6
      • Do this until you run out of pairs
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4
Q

Negative Factors

A
  • Technically divisors and factors could be negative
    • -3, -4, -6, -2, -1, -12, are all factors of -12 and 12
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5
Q

Divisibility Rule for 2

A
  • all even numbers are divisible by 2
  • just look at the last digit
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6
Q

Divisibility Rule for 5

A
  • if the last digit is 5 or 0 the number is divisible by 5
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7
Q

Divisibility Rule for 4

A
  • if the last two digits of a number, taken as a single number, are divisible by 4 then the entire number is divisible by 4
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8
Q

Divisibility Rule for 3

A
  • add all individual digits together
  • if their sum is divisible by 3 then then the number is divisible by 3
  • 234,837 -> 2+3+4 = 9, 8+3+7 = 18 -> 18+9 = 27 (YES!)
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9
Q

Divisibility Rule for 6

A
  • the number must be…
    • divisible by 2 (check last digit)
    • divisible by 3 (add digits together)
    • 1296 -> 1+2+9+6 = 18 (YES!)
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10
Q

Divisibility Rule for 9

A
  • add all of the digits together
  • if the sum is divisible by 9 then then number is divisible by 9
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11
Q

Multiple

A
  • a number found by multiplying one of its factors
  • 75 and 1250 are multiples of 5
  • 63 and 888 are multiples of 3
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12
Q

Multiple Idea (General)

A
  • If P and Q are multiples of r, then the sum, difference, or product of P and Q are multiples r
  • Multiples of 8
    • 24 + 80 = 108
    • 80 - 24 = 56
    • 24 * 80 = 1920
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13
Q

Multiple Idea - Multiples of 1

A
  • every integer is a multiple of 1
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14
Q

Multiple Idea - Integers

A
  • to find multiples you multiply the factor times integers
  • 3 * {1,2,3,4,5}
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15
Q

Multiple Idea - Base Addition & Subtraction

A
  • you can simply add or subtract the integer to get the multiple
  • 2401 + 7 = 2408
  • 2408 + 7 = 2415
  • 2401 - 7 = 2393
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16
Q

Multiple Idea - Multiple Addition & Subtraction

A
  • if P and Q are multiples of r, then (P+Q) and (P-Q) are multiples of r
    • 700 + 49 = 749
    • 749 +49 = 798
    • 700 - 49 = 651
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17
Q

Multiple Idea - Multiples of Multiples

A
  • if P is a multiple of r, then any multiple of P is a multiple of r
    • 52 is a multiple of 13 so any multiple of 52 is a multiple of 13
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18
Q

Multiple Idea - Products of Multiples

A
  • if P and Q are multiples of r then the product of P*Q is a multiple of r
  • 24 and 80 are both multiples of 8, so are the following
    • 24 + 80 = 104
    • 80 - 24 = 56
    • 24*80 = 1,920
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19
Q

What is a Prime Number

A
  • an integer with only two factors
  • prime numbers are the building blocks of all positive numbers
20
Q

Prime Number Facts

A
  • 1 is NOT a prime number, since it has only one factor
  • 2 is the only even prime number
21
Q

Prime Number Test

A
  • test whether a number is divisible by any prime less than 10
    • 2, 3, 5, 7
    • if not then it is a prime
  • for larger numbers find the closest multiples of these numbers then add another to see if it fits
    • For 57
      • 7*8 + 7= 63 (NO)
      • 5*11 + 5 = 60 (NO)
      • 3*20 - 3 = 57 (YES)
22
Q

Prime Factorization

A
  • the prime numbers that multiply to create an integer
  • the DNA of a number, revealing all of its ingredients
  • an extremely important skill
  • Examples
    • 9 = 3*3
    • 10 = 2*5
    • 12 = 3*4 = 3*2*2
    • 24 = 8*3 = 2*4*3 = 2*2*2*3 = 23*3
    • 100 = 10*10 = 2*5*2*5 = 22*5
23
Q

Prime Factorization (multiples)

A
  • if r is a prime factor of Q, then ALL prime factors of r are prime factors of Q
24
Q

Large Number Factoring

A
  • find the prime factorization: 8400 = 24 * 3 * 52 * 7
  • make a list of the exponents of the prime factors: {4,1,2,3}
  • add 1 to every number on the list: {5,2,3,4}
  • multiply those numbers together: 5*2*3*4 = 120
25
Q

Finding Odd Number of Factors

A
  • prime factorize: 21600 = 25 * 33 * 52
  • list the exponents of the odd prime factors bases: {3, 2}
  • add 1 to each: {4, 3}
  • multiply them together: 4 * 3 = 12
26
Q

Why the procedure for finding the number of factors works

A
  • 21600 = 25 * 33 * 52
  • if factor “F” is “built” on these prime factors, how many powers of 2 could F contain? It may contain none (3*5 = 15). Any from 0 to 5, which is 6 possibilities {0,1,2,3,4,5}
    • Possibilities for base factor 3 {0,1,2,3}
    • Possibilities for 5 {0,1,2}
  • think of it as 3 slots to build a factor
    • Slot 1: factors for 25: 0 - 5 = 6
    • Slot 2: factors for 33: 0 - 3 = 4
    • Slot 3: factors for 52: 0 - 2 = 3
27
Q

Squares of Integers

A
  • perfect squares
  • exponents of the prime factors of a square are all even
  • if all exponents are even in a prime factorization then it is a perfect square
28
Q

Counting Factors in a Perfect Square

A
  • all exponents are even
  • when 1 is added for multiplying all exponents become odd
  • so a perfect square will always have an odd number of factors
29
Q

Multiplying Odd Numbers

A
  • the product of an all odd number problem is an odd number
  • 3*3 = 9
  • 5*7 = 35
  • 9*13 = 117
30
Q

Greatest Common Factor

A
  • also called the Greatest Common Divisor
  • the largest factor that two numbers share: 24 & 40 = 8
31
Q

Greatest Common Factor for Large Numbers

A
  • 360 and 800
  • prime factorize:
    • ​360: 23 * 32 * 5
    • 800: 25 * 52
  • find the biggest factors shared and multiply
    • 23 * 5 = 40
      *
32
Q

Least Common Multiple

A
  • lowest multiple that two numbers share
  • 8 & 12 = 24
33
Q

Find the Least Common Multiple

A
  • prime factorize: 24 & 32
    • 24: 23 * 3
    • 32: 25
  • find the GCF: 23
  • multiply the GCF * the smallest remaining factor
    • 24: 23 * 3
    • 32: 23 * 4
  • multiply each of these factors together: 23 * 3 * 4 = 96
34
Q

Least Common Multiple Symbolic Representation

A
  • M and N are the two numbers and G is the GCF
  • N/G = A
  • M/G = B
  • LCM = A * B * G
35
Q

Why learn the Least Common Multiple

A
  • the test will ask for the LCM in hidden ways!
    • Hot dogs are 8 to a pack and buns are 12 to a pack. What’s the least amount of hot dogs to put together with no leftovers? 24
  • it is also important for adding/subtracting fractions. the LCM is the LCD
36
Q

Factors and LCM

A
  • if A is a factor of R then the LCM of A and R must be R
    • the LCM of 8 and 24 is 24
  • if A and B have no factors greater than 1 then the LCM is A*B
    • the LCM of 7 and 15 is 7*15 = 105
37
Q

GCD LCM Formula

A
  • take all factors of one multiple and multiply by the remaining factors of the other
    • 18 and 24
      • 18: 32 * 2 = 3 * 3 * 2
      • 24: 23 * 3 = 2 * 2 * 2* 3 * 3
      • 3 * 3 * 2 * 2 * 2 = 72
38
Q

Zero

A
  • Zero is neither positive or negative
  • Zero is even
39
Q

Odd - Addition Subtraction

A
  • Add/Subtract unlikes equals odd
    • 13 + 8 = 21
    • 13 - 8 = 5
40
Q

Even - Addition Subtraction

A
  • Add/Subtract likes is even
    • 5+7 = 12
    • 7-5 = 2
41
Q

Odd Even - Even Factor

A
  • If there is an even factor in multiplication then the number must be even
    • 3 * 5 * 4 = 15 * 4 = 60
42
Q

Odd Even - Odd Factors

A
  • For a number to be odd all factors must be odd
    • 3 * 5 * 7 = 105
43
Q

Odd Even - Division

A
  • no hard rules for division
  • the quotient could be even, odd, or a non-integer
44
Q

Integer Caution!

A
  • never assume a number is an integer
  • it could be a fraction in which case it is neither odd nor even
45
Q

Odd Even - Fractions

A
  • fractions are neither odd nor even

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