GMAT Quant Chapter 3: Properties of Numbers Flashcards
For the GMAT, is 0 considered to be positive or negative?
Neither.
The GMAT does not consider 0 to be positive or negative.
What is the square root of 0?
Is 0 even or odd?
0 is the only number that …?
0
Even
is equal to its opposite.
What is the number with only 1 factor?
What is the first prime number?
2.
What do even exponents always produce?
Positive results.
How can we determine all the factors of a number?
Start at 1 and list all of the factors.
What are the first 25 prime numbers?
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
Define a composite number.
How can we express the factors of this type of number?
Any number that is not prime.
As a product of its prime factors.
How do we find the number of factors for a given number?
Step I. Prime Factorize
Step II. +1 to the value of each exponent of the PFs.
Step III. Multiply all of the (PFs +1) together
What is the difference between a unique prime factor and a prime factor?
A unique prime factor is the only prime factor for a given number.
A prime factor is any number that is only divisible by 1 and itself.
How do we calculate the Lowest Common Multiple (LCM) of two numbers?
- Prime Factorize each integer
- If any PFs are repeated, take only the largest exponent
- Take all non-repeated PFs
- Multiply the values from Steps 2 & 3
How do we find the LCM for more than 2 integers?
- PF
- Find repeated PFs with highest exponent (PF shared by at least 2 numbers)
- Take all non-repeated PFs
- Multiply steps 2 & 3 values
How do we calculate the LCM if two integers do not share PFs?
Multiply the integers together.
How do we find the Greatest Common Factor (GCF)?
- PF each number
- Find repeated PFs
- take smallest exponents of repeated PFs
- Multiply numbers from step 3
How do we calculate the GCF if there are no repeated PFs?
take only those with the smallest exponent
What is the GCF if a set of integers has no prime factors?
1.
If we have two integers x and y, and we know the LCM of one and the GCF of another, what can we calculate?
LCM x GCF of two seperate integers = the product of those two integers.
What is the result when we multiply all of the unique prime factors?
The LCM.
To check whether a certain number is divisible by another, what must we do?
Prime factorize to see if the denominator cancels out with the numerator.
If x is divisible by y, then what must be true about the divisibility of x?
x must be divisible by all of the factors of y
If z is divisible by x and y, what must be true about the divisibility of z?
Z must be divisible by the LCM of x and y but not higher multiples of the LCM.
Is 0 divisible?
0 is divisible by any number other than itself.
How do we know if a number is divisible by 11?
If the sum of the odd numbered placed digits minus the sum of the even placed digits is divisible by 11
How do we know if a number is divisible by 12?
If it is both divisible by 3 and 4.
What will the product of any n consecutive integers be divisible by?
n!
what is the algebraic form of factored and non-factored consecutive integers?
n(n -1)
n(n+1)
n squared – n
n squared + n
What is the product of n consecutive even integers divisible by?
2 to the power n
*
n!
What is the division formula?
Dividend = Divisor x Quotient + remainder
How do we calculate the remainder for two integers multiplied together?
calculate remainders for each integer individually, multiply the remainders together,
divide result so that remainder is less than the original divisor
Can remainders be added and subtracted?
Yes if we make sure the result is not larger than the divisor
How do we calculate the number of trailing 0s for any given number?
PF and the number of (5x2) pairs is the number of trailing 0s.
What do we know about a factorial greater than or equal to 5?
0 will always be its units digit.
How do we calculate the total number of digits for a calculation involving massive exponents?
- PF the numbers.
- Count the number of (5x2) pairs. Each pair = one 0
- Collect the number of unpaired 5 or 2s & any other nonzero PFs. Multiply them all together. Count number of digits in product.
- Sum number of digits from steps 2 & 3
How can we quickly determine how many leading 0s a decimal has?
If X is an integer with k digits & x is not a power of 10. Then 1/x will have (k-2) leading 0s in its decimal form.
What is the product of any consecutive integers always divisible by?
Any of the integers in the set or by any of the factor combinations of the numbers
How do you determine the number of a given prime number in a factorial?
Divide the integer by the PF in its progressive powers.
e.g.
integer / 3 = x
integer / 9 = y
integer / 27 = z
integer / 81 = 0
The sum of x,y, and z is the number of PFs in that factorial.
What are the first 9 perfect cubes?
0
1
8
27
64
125
216
343
512
Define terminating decimal.
A number with a finite number of digits to the right of the decimal point.
How do we know if a fraction produces a terminating decimal?
If, when factorized, the denominator contains only 2s or 5s then the decimal will terminate.
A factorized denominator with any other prime factors will produce a decimal that does not terminate.
Name the pattern of the units digits when a number is divided by the integers 1 through 9.
2: 2-4-8-6 Repeat
3: 3-9-7-1 Repeat
4: 4-6 Repeat
5: All end in 5
6: All end in 6
7: 7-9-3-1 Repeat
8: 8-4-2-6 Repeat
9: 9-1 Repeat
What do we know about 2 consecutive integers and their PFs?
They will never share the same prime factors.
What is the GCF of two consecutive integers?
1
