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GMAT - Quant > Chapter 4 - Properties Of Numbers > Flashcards

Chapter 4 - Properties Of Numbers Flashcards

1
Q

When will the sum or difference of two integers be even?

A

Only when both numbers are even or both numbers are odd

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2
Q

When will the product of two integers be even?

A

The product of two integers is always even if one of the integers is even

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3
Q

When will the product of two integers be odd?

A

The product of any two odd numbers will always be odd

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4
Q

When an even number is divisible by an odd number, will the result be even or odd?

A

The result will be even

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5
Q

When an odd number is divisible by an odd number, will the result be even or odd?

A

The result will be odd

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6
Q

What is a factor?

A

Given two numbers, x and y, if y divides evenly into x, we say y is a factor of x.

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7
Q

What is a multiple for a given number?

A

A multiple of a number is the product of that number with any integer.

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8
Q

What are the prime numbers between 1 and 100?

A

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97

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9
Q

What is the formula to find the sum of all factors of a number?

A

After identifying the unique prime factors, where the prime factors are axbycz:

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10
Q

What is the process for finding the LCM of two numbers?

A
  • Step 1: Find the prime factorization of each integer. In other words, prime factorize each number and put the factors in exponent form.
  • Step 2: If there are repeated prime factors in the set, take only those with the largest exponent. If you have two prime factors with the same exponent, take that number only once.
  • If we had 32 and 33, we would take 33.
  • Step 3: Of what is left, take all the non-repeated prime factors of the integers.
  • Step 4: Multiply together the numbers you found in steps 2 and 3
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11
Q

What is the process for finding the GCF of two numbers?

A
  • Step 1: Find the prime factorization of each number
  • Step 2: Identify repeated prime factors among the numbers
  • Step 3: Among the repeated prime factors, take only those with the smallest exponents. If no repeated prime factors are found, the GCF is 1.
    • Important: you’re only taking the repeated prime factors with the lowest exponents, if they’re not repeated ignore them
  • Step 4: Multiply the numbers you found in steps 2 and 3; this product is the GCF
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12
Q

How can we solve repeating pattern questions? (ie - event A occurs every 18 seconds, event B occurs every 36 seconds)

A

Find the LCM of the two timings, and that will tell you when they will align again.

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13
Q

When does an even division occur?

A

An even division occurs when the numerator of a positive fraction is a multiple of the denominator.

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14
Q

What are the divisibility rules for 0?

A

No number is divisible by zero

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15
Q

What are the divisibility rules for 1?

A

All numbers are divisible by 1, and the result is the same as the dividend

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16
Q

What is the divisibility rule for 3?

A

A number is divisible by 3 if the sum of its digits is a multiple of 3.

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17
Q

What are the divisibility rules for 4?

A

If the last two digits are divisibly by 4, then the entire number is divisible by 4

Don’t forget: 00 is divisible by 4

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18
Q

What are the divisibilty rules for 5?

A

A number is divisible by 5 if it ends in 0 or 5

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19
Q

What is the divisibility rule for 6?

A

A number is divisible by 6 if it’s an even number whose digits sum to a multiple of 3 (and therefore has factors 2 and 3, which are also the factors of 6).

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20
Q

What is the divisibility rule for 7?

A

It’s too complicated! Just do the long division.

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21
Q

What is the divisibilty rule for 8?

A

If the number is even, divide the last three digits by 8. If there is no remainder, the number is divisible by 8.

22
Q

What is the divisibility rule for 9?

A

If the sum of a numbers digits is a multiple of 9, the number is divisible by 9

23
Q

What is the divisibility rule for 10?

A

If the ones digit is a 0, then it’s divisible by 10

24
Q

What is the divisibility rule for 11?

A

A number is divisible by 11 if the sum of the odd numbered digits (the 1st, 3rd, 5th, etc. digits) minus the sum of the even numbered digits is divisible by 11.

25
Q

What is the divisibility rule for 12?

A

If a number is divisible by both 3 and 4, it is also divisible by 12

26
Q

How do you determine the number of trailing zeros in a number?

A

Prime factorize, and count the number of (5x2) pairs

27
Q

How do you determine the number of digits in an integer?

A
  • Step 1: Prime factorize the number
  • Step 2: Count the number of (5x2) pairs to get the number of trailing zeros
  • Step 3: Collect the number of unpaired 5s or 2s, along with other nonzero prime factors (if any) and multiply them together. Count the number of digits in this product.
  • Step 4: Sum the number of digits from steps 2 and 3.
28
Q

How do you determine the number of leading zeros in a fraction?

A
  • First, express the fraction in the form 1/x, where x is an integer.
  • If 1 < x ≤ 10, there are no leading zeros
  • If 10 < x ≤ 100, there’s one leading zero
  • If 100 < x ≤ 1,000, there will be two leading zeros, etc.
29
Q

How do you determine the number of leading zeros in a fraction 1/X (where X is not an integer)?

A
  • If x is not an integer, first express the fraction in the form of 1/x.
  • For example, if we are given the fraction 2/25, we express it as 1/12.5
  • If 1 < x < 10, there will be no leading zeros
  • If 10 < x < 100, there will be one leading zero
  • If 100 < x < 1,000, there will be two leading zeros
  • If 10n < x < 10n+1 (where n is a non-negative integer), then there are n leading zeros.
30
Q

What is the shortcut for determining the number of primes in a factorial? (The number of 3s, for example)

A
  • Step 1: Divide the number of the factorial by the prime you’re attempting to determine, increasing the exponents: 31, 32, 33… 3k and ignore any remainders. Stop dividing when you get a quotient of zero.
  • Step 2: Add the quotients from 31, 32, 33…3k. The sum of the quotients represents the number of a given prime in the factorial.
31
Q

Perfect squares will always end in which digits?

A

0, 1, 4, 5, 6 or 9

32
Q

Which fractions/division problems produce terminating decimals?

A

Any denominator whose prime factorization contains only 2s, 5s or both produces decimals that terminate.

33
Q

What is the units digit pattern for powers of 2?

A

2-4-8-6

34
Q

What is the units digit pattern for powers of 3?

A

3-9-7-1

35
Q

What is the units digit pattern for powers of 4?

A

4-6

(odd powers are 4, even powers are 6)

36
Q

What is the units digit pattern for powers of 5?

A

All powers of 5 end in 5

37
Q

What is the units digit pattern for powers of 6?

A

All powers of 6 end in 6

38
Q

What is the units digit pattern for powers of 7?

A

7-9-3-1

39
Q

What is the units digit pattern for powers of 8?

A

8-4-2-6

40
Q

What is the units digit pattern for powers of 9?

A

9-1

(Odd powers end in 9, even powers end in 1)

41
Q

What is the value of 1!?

A

1

42
Q

What is the value of 2!?

A

2

43
Q

What is the value of 3!?

A

6

44
Q

What is the value of 4!?

A

24

45
Q

What is the value of 5!?

A

120

46
Q

What is the value of 6!?

A

720

47
Q

What is the value of 7!?

A

5,040

48
Q

What is the value of 8!?

A

40,320

49
Q

What is the value of 9!?

A

362,880

50
Q

What is the value of 10!?

A

3,628,800

51
Q

What is the formula for division?

A
52
Q

What is the formula for determining the number of factors of a number?

A

n = (e1 + 1)(e2 + 1)(e3 + 1)…(en + 1) where e is the exponent of each prime factor.

GMAT - Quant (6 decks)

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