GMAT Number Properties 2

Counting Total Factors and how many factors number 30 has?

To find the total number of factors in a large number: 1. Break a number into it's prime factorization. 2. Add a one to each prime's exponent. (remember that numbers that don't have a factor have an implicit factor of 1) 3. Multiply all the results of #2 together to get the number of different factors. 30 has 8 factors.

How many multiples of 2 are from 20-40

11 40-20=20 20/2=10 and add 1 (the first number belongs to the set) so 11.

divisible by 5

integer ends in 0 or 5

How to get a specific remainder

- add the desired remainder to a multiple of the divisor - ex: need a number that leaves a remainder of 4 after dividing by 7; (7*2) + 4 = 18

GCF if no primes in common

1 (not zero!)

If s and t are positive integers such that s/t = 64.12, which of the following could be the remainder when s is divided by t ? (A) 2 (B) 4 (C) 8 (D) 20 (E) 45

12/100 3/25 Divisor must be a multiple of 25 and remainder a multiple of 3 45 is only mult. of 3 so E

5!

120

Prime numbers between 1 and 50

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47

For any positive integer n, the sum of the first n positive integers equals n(n+1)/2. What is the sum of all the even integers between 99 and 301?A. 10,100B. 20,200C. 22,650D. 40,200E. 45,150

20200 C When calculating evenly spaced sets of multiples and first and last numbers are not multiples of the number asked. It is best to find the closest multiples within range and use them in subtraction. You can use divisibility rules for this. Evenly spaced set of even integers begins from 100 and continues to 300. 300-100 = 200 200/2=100 100+1=101 Average of this set is 99+301=400 400/2=200 To get the sum of evenly spaced sets AVG*number of items in the set. So 200*101=20200

4!

24

How many integers are from 10-541?

532 When the first item is part of the series you must include it. So 541-10=531 and then add 1 so 532.

3!

6

6!

720

Multiple +/- multiple

=multiple

Multiple +/- non-multiple

=non-multiple

When are all evenly spaced sets determined

All evenly spaced set are fully defined if following are defined: i. First number or last number ii. The number of items in the set iii. The increment

What is the rule for odd/even in addition or subtraction

Both are same type (o+o) or (e+e) then = even If one is different (o+e) then = odd

If x and y are positive integers such that y is a multiple of 5 and 3x + 4y = 200, then x must be a multiple of which of the following? A) 3 B) 6 C) 7 D) 8 E) 10

Concept multiples and divisibility, alg.

Prime factorization of perfect squares

Contains only even powers of primes

What is the lowest positive integer that is divisible by each of the integers 1 through 7, inclusive?(A) 420(B) 840(C) 1,260(D) 2,520(E) 5,040

Don't get fooled into counting 7!. You need the lcm of 1-7. So 2,3,4=(2^2),5,6=(3*2),7 So we need 2^2*3*5*7=420 so A

Greatest Common Factor (GCF) & Least Common Multiple (LCM)

GCF: the largest divisor of 2+ integers LCM: the smallest multiple of 2+ integers - use a venn diagram to find GCF and LCM 1. Factor both numbers into primes 2. Place common factors in the shared area (incl. copies) 3. Place non-common factors in the sides 4. GCF = product of primes in the shared area 5. LCM = product of all primes in the diagram - if no primes are in common, the GCF is 1 and the LCM is the product of the two numbers

If k is a positive integer, what is the remainder when (k + 2)(k^3 - k) is divided by 6? A. 0 B. 1 C. 2 D. 3 E. 4

Good case for plugging in numbers since only one can be the answer. k=1 =0 0 leaves a remainder of 6 when divided by 6 so = A

Factor Foundation Rule

If a is a factor of b and b is a factor of c, then a is a factor of c ex: 2 is a factor of 4 and 4 is a factor of 8; so 2 is a factor of 8 --aka: any integer is divisible by all factors of its factors

What is the rule for o/e in multiplication

If they are same (o*o) then they will be of the same type. So o*o = o e*e=e If different then the product will be even.

Divisibility & Addition/Subtraction

If you add or subtract Multiples of N, the result is a multiple of N. - N is the divisor of x and y, then N is a divisor of x + y - ex: 35 + 21 = 56 If you add or subtract a multiple of k and a non-multiple of k the sum or difference will not be a divisible by k.

How to get the sum of an evenly spaced set

Just calculate the mean and multiply by the number of items. As mean is the sum/n of items.

An evenly spaced set has 60 items, is the average an integer?

No

Is 3^4 a multiple of 4?

No, even though 3 is multplied 4 times by itself it will not be a multiple of 4 as o*o*o*o=odd. You may be mixing things up with multiplication of four consecutive integers. Four consecutive integers will be a multiple of 4 and 3 and 2 and ofc 1. n! will be always divisible by n and anything n<. This is because how multiples are located on the number line. eg. every third number is a multiple of 3. So 3! would be multiplied by a multiple of 3 and due factor foundation rule we know that this sum must be then divisible by 3.

Name the exponent rules for odds and evens

Odd with any positive exponent = Odd o*o Even with any positive integer = Even e*e

The sum of prime numbers that are greater than 60 but less than 70 is(A) 67(B) 128(C) 191(D) 197(E) 260

Prime interval test Approach: 1. Eliminate even numbers, numbers ending in 5 and numbers ending in 0. Numbers left 61,63,67,69 2. Use divisibility rules. 63 div 3, 69 div 3 Left with 61 and 67. 3. Check for divisibility by running through multiples of 7. 56 = multiple of 7 63, 70. 61+67=128 (c)

Consecutive Multiples

Special cases of evenly spaced sets: all values in the set are multiples of the increment - ex: 12,16,20,24 - increase by 4s, ea. element a multiple of 4 - these sets must be composed of integers

When positive integer x is divided by positive integer y, the remainder is 9. If x/y = 96.12, what is the value of y? (A) 96 (B) 75 (C) 48 (D) 25 (E) 12

Take the remainder Turn it into a fraction Look for multiples

How to get the average of a set if you only know the first and last items

Take their average

What the arithmetic mean of evenly spaced sets equal

The median because the numbers grow at a steady pace half of the numbers will be below the average and half will be above

What the amount of even numbers in a product tells us about divisibility

The number of even numbers tells us what power of 2 the product is divisible by. E.g 20*12=240 is at least divisible by 4.

What is the greatest number of identical bouquets that can be made out of 21 white and 91 red tulips if no flowers are to be left out? (Two bouquets are identical whenever the number of red tulips in the two bouquets is equal and the number of white tulips in the two bouquets is equal.) A. 3 B. 4 C. 5 D. 6 E. 7

This is a hidden GCF. The identical clause identifies that we are looking GCF. Another way to ask this is that the ratio of flowers must be same in each boquet. Also no leftovers is a indicator of hidden gcf. And "what is the greatest number of..." Number that will divide both 91 and 21. This will give us identical boquets. GCF of 91 and 21 is 7. 13*7 and 3*7

If n = 4p, where p is a prime number greater than 2, how many different positive even divisors does n have, including n? (A) Two (B) Three (C) Four (D) Six (E) Eight

This is type prime greater than. In these picking numbers is very beneficial because there can be only one right ans. So lets pick 3 as our prime. 3*4=12 We cannot just calculate how many even numbers there is 2-12 as not every even will be a divisor of 12. Lets list of div of 12... 2,4,6,12 eliminate 3 bc not even so 4 (C)

When positive integer A is divided by positive integer B, the result is 4.35. Which of the following could be the remainder when A is divided by 8? (A) 13 (B) 14 (C) 15 (D) 16 (E) 17

We know that remainders in fraction form must be integers. So let's turn 0,35 into frac 35/100 7/20 This tells us that to get 0,35 we need a divisor that is a multiple of 20 and a remainder that is a multiple of 7 to keep this ratio. 14 is the only mult. of 7 so 14 is ans. If 14 is remainder divisor must be 40. and A must be 174.

Disguised Positive & Negative Questions

Whenever you see inequalities with zero on either side of the inequality, you should consider testing positive/negative cases to help solve the problem.

An evenly spaced set has 61 items, is the average an integer?

Yes

divisible by 6

integer divisible by both 2 and 3

divisible by 8

integer is divisible by 2 three times or if the last 3 digits are divisible by 8

divisible by 10

integers ending in 0

divisible by 4

last two digits are divisible by 4 or divisible by 2 twice

Odd number of negative signs (multiplication/division)

negative

Even number of negative signs (multiplication/division)

positive

LCM if no primes in common

product

divisible by 7

rare, but need long division

divisible by 3

sum of integer's digits is divisible by 3

divisible by 9

sum of the integer's digits is divisible by 9

Remainder formula and properties of remainders

x = Q*N + R - x= dividend - Q=quotient - N=divisor - R=remainder *all must be integers The remainder of any number must be non-negative and smaller than the divisor. When you divide by a positive integer N, there are N possible remainders.