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34 Cards in this Set
- Front
- Back
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Divisibility & Primes
An integer is divisible by 3 if.. |
sum of digits is divisible by 3
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An integer is divisible by 4 if ..
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divisible by 2 twice
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An integer is divisible by 5 if ..
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ends in 5 or 0
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An integer is divisible by 6 if ..
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divisible by 2 and 3
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An integer is divisible by 8 if ..
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divisible by 2 3 times
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An integer is divisible by 9 if ..
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sum of digits is divisible by 9
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Every number has limited factors but infinite multiples
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If 2 numbers have common divisor, their sum and difference retain the divisor as well
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Prime Number - factors 1 & itself
2 is on the only even prime Memorize :2,3,5,7,11,13,17,19,23,29 |
1 is a non-prime factor of all integers
Factor Foundation Rule: If 72 is divisible by 12, then it is divisible by all factors of 12 If 12 is a factor of 72, then all factors of 12 are factors of 72 |
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Prime Product
Is 27 a factor of 72? 72 - 2,2,2,3,3 27=3x3x3 we cannot make 27 from factors of 72. so not divisible |
Using prime factors, we can find all possible prime products and factors of a number
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GCF
30-2,3,5 24-2,2,2,3 lower power product of prime factors. 2x3=6 |
LCM
30-2,3,5 24-2,2,2,3 higher power product of prime factors 2^3 x 3 x 5 = 120 |
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Problem 2
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Problem 5
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Odds & Evens
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Rule1:Addition/Subtraction
Add 2 odds or add 2 evens, EVEN you shall see But add an odd with an even, how ODD it will be |
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Rule 2:Multiplication
One even, ans is even converse is also true no even->ans is odd |
Rule 2: Division
no guaranteed outcome as numerator can be < denominator |
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Sum of 2 primes
If sum of 2 primes is odd, then one of them is 2, otherwise sum of 2 primes is even |
Positive & Negative
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If ans is +ve or -ve, check for all 4 possibilities (++),(+-).(-+),(--)
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Consecutive Integers
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no. of integers between x & y inclusive is (y-x) + 1
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Sum of consecutive integers from a to b
((a+b)/2)x(b-a+1)->no of terms =(n/2)x(a+b) |
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Avg term in consecutive set
If [a,b] given, avg = (a+b)/2 if sum & n given -> sum/n |
Avg of odd no. of consecutive integers is always integer
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Avg. of even no of consecutive integers is never an integer as there is no middle no
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Special Products
Product of 'n' set of consecutive integers is divisible by n |
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Special Sums
1+2+3=6. Divisible by 3 For any set of consecutive integers with odd no of terms, sum of all integers is always multiple of no of terms Not For even number of terms |
if x^3-x=1, x is even, is p divisible by 4?
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(x-1)(x)(x+1)
One of these factors would be even. So not necessary |
For consecutive multiples of a number, one of them will be even
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EXPONENTS
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For +ve bases > 1, the greater the exponent, the faster the rate of increase
5^2=25, 5^3=125. Increased by 100 |
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If x^6=x^7=x^15, then x is either 0 or 1
If x^6=x^8=x^10, x could be 0,1 or -1 |
For fractions 3/4 >(3/4)^2>(3/4)^3
4^6=4096 |
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2^4x2^5/2^7-2^4
First multiply, divide then subtract |
x^3=x^15, then x=0,1,-1
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if x^3<x^2, then x is a fraction or -ve number
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ROOTS
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sqrt(4) = x, x=2.
sqrt of a number is always non-negative |
if sqrt(x) is integer, then x is a perfect square
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sqrt(52)=?
sqrt(49)<sqrt(52)<sqrt(64) since 52 is closer to 49, sqrt(52) is approx 7.2 |
1.4^2 = 2 approx
1.7^2 = 3 approx PEMDAS |
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DATA SUFFICIENCY STRATEGY
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Rephrasing
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if p is an integer, is p divisible by 18?
1)5p/18 is an int 2)6p/18 is an int |
18=2x3x3
Are there 2 3's and a 2 in p's prime box? 1)5p/18 5 2x3x3 SUFFICIENT 2)6p/18 (2x3) 3 NOT SUFFICIENT |
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if x is a +ve integer, is x^3-3x^2+2x div by 4?
1.x=4y+4, y is int 2.x=2z+2, z is int |
Rephrase by factoring
x(x^2-3x+2)=x(x-1)(x-2) 3 consecutive integers.Is product of 3 consecutive int div by 4? only possible if two of them are even. x and x-1 are even.or x-1 is div by 4 1.x is even - sufficient 2.x is even - sufficient |
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Types of Ds problems -Yes/No vs Value
Value if info leads to single answer, then sufficient. If info leads to more than one answer, not sufficient Yes/No If ans is yes or no, sufficient If ansis maybe, insufficient |
Testing Numbers
Value If n is int, 1<=n^3<=100, what is n 1.n=2k+1,k int 2. n is prime Sol-Limited values for n = 1,2,3,4 1.1,3 possible choices - not sufficient 2.2,3 possible choices - not sufficient Together they are sufficient |
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Yes/No
Is positive int n div by 108? 1.n is div by 12 2.n is div by 9 Sol-108 = 2x2x3x3x3 1.2x2x3 , Not sufficient 2.3x3, Not sufficient combined 2x2x3x3. not sufficient |
Testing Smart Numbers
->Try to prove stmt is insufficient for fractions,negatives and zeros unless anything mentioned |
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Product of 3 consecutive integers for 990, think of 1000 -10x10x10
so 990=9x10x11. |
Stmts will never contradict each other.
Whenever contradiction in stmt answers, you have made mistake |
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in divisibility remainder problems, when remainder mentioned r, start picking numbers for denominator/divisor from r+1
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for (x+y)/y
remainder is same as x/y |
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sum of first n odd numbers = n^2
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sum of first n even numbers = n(n+1)
For 1 to n if n is odd, then there are (n-1)/2 even integers |
