3. Properties of Numbers

Find the GCF Example: What is the GCF of 60 and 72?

1) prime factorize each integer: 60 = 3^1 x 2^2 x 5^1 and 72 = 2^3 x 3^2 2) pick out smallest exponents ONLY of shared prime factors: 2^2, 3^1 3) multiply together 2^2 x 3^1 = 12

Converting a remainder from decimal form to fraction form Example: x/y = 9.48

1) remainder is not necessarily 48 2) reduce decimal to most reduced fraction: 12/25 3) actual remainder must be multiple of 12

Find the LCM Example: What is the LCM of 24 and 60?

1) prime factorize each integer: 24 = 3^1 x 2^3 and 60 = 3^1 x 2^2 x 5^1 2) pick out high exponents of unique factors: 2^3, 3^1, 5^1 3) multiply together 2^3 x 3^1 x 5^1 = 120

Formula to determine total number of factors of a number Example: What is the total number of positive numbers that will divide into 240?

-(e_1+1)x(e_2+1)x(e_3+1) -e=exponent of prime factor 1) Prime factorize 240: 2^4 x 3^1 x 5^1 2) Add one to each exponent: (4+1) x (1+1) x (1+1) 3) Finish: 5 x 2 x 2 = 20

GCF of 2 consecutive factors Example: If X and Y are positive, consecutive numbers, what is the GCF?

-Consecutive numbers never share factors Example: 1

Terminating decimals Which decimal does not terminate? 1) 7/80 2) 5/60 2) 2/20

-Decimal will only terminate if only containing 2's, 5's, or both 1) 7/80 = 7/(2^4*5) terminates 2) 5/60 = 5/(2^2*3) does not terminate, correct 3) 2/20 = 1/(5*2) terminates

-Even/odd rules for addition/subtraction -Even/odd rules for multiplication -Even/odd rules for division

-If they are the same, even, otherwise odd -If there is a even number, even, otherwise odd -even/odd=even -odd/odd=odd -even/even=even OR odd -odd/even = no solutions

Leading zeros in number 1/X Example 1: Leading zeros in 1/(5^5x2^11) Example 2: Leading zeros in 1/(5^5x2^5)

-K-1 or K-2 for perfect power of 10 -K = digits in X Example 1: 1) 5 5's matched with 5 2's equals 5 trailing zeros 2) 6 2's left over = 2^6 = 64 3) 64 + 5 zeros = 7 total digits 4) k - 1: 7 - 1 = 6 leading zeros Example 2: 1) 5 5's matched with 5 2's equals 5 trailing zeros 2) 1st digit in front of 5 trailing zeros is 1 3) 1 + 5 zeros = 6 total digits 4) k - 2 (perfect power of 10): 6 - 2 = 4 leading zeros

If Z is divisibly by both X and Y, Z must also be divisible by... DS Question: If A and B are positive integers, and A is a multiple of 12, is AB divisible by 75? 1) A is divisible by 60 2) B is divisible by 35

-LCM of X and Y Stim analysis: A is multiple of 12, meaning A/2^2x*3^1 is an integer. If prime factorize 75 to 3^1x5^2, then need at least 2 5's 1) prime factorize 60: 2^2x3^1x5^1, insufficient 2) prime factorize 35: 5^1x7^1, insufficient Both) multiple 60*35 gives to 5's. AB/60*35 SUFFICIENT

LCM and Unique Prime Factors DS: How many unique prime factors are in the product of positive integers S & T? 1) The LCM of S and T is 120 2) The GCF of S and T is 6

-LCM provides all unique prime factors 1) Sufficient, 120 = 3^1x2^3x5^1 2) Insufficient

Odd number of factors Example: How many numbers between 1 and 1000, inclusive have an odd number of factors?

-Only perfect squares have odd number of factors (2^2 has factors of 1,2,4) 1) Only perfect squares have odd factors 1^2, 2^2,3^2... 2) 31^2=961<1000, 32^2=1024>1000 3) 31

Perfect square must... DS Question: Is positive integer X a perfect square? 1) X = t^n, where t is positive integer and n is odd 2) X^0.5 = k, where k is positive integer

-Perfect squares must have ALL even exponents 1) If X = 2^3, not perfect. If X = 1^3, perfect. INSUFFICIENT 2) Since k is positive, SUFFICIENT

Prime number attributes DS: If X is a positive number, then is X a prime number? 1) X is a multiple of a prime number 2) X can be represented as the product of two integers

-Prime numbers ARE multiples of themselves 1) If X=2 then Yes (2 is prime and multiple of prime number 2, itself). If X=4 then No. Insufficient 2) 1*2=2 then Yes. 1*4=4 then No. Insufficient Both: From answers above, still Insufficient

Multiplying excess remainders Example: If X = 500x600x700, what is the remainder when X is divided by 8?

-Remainders can be multiplied but must correct for excess remainders -Same process can be repeated for adding subtracting numbers and finding remainder (can't have negative remainder though, must add back divisor until positive) 1) 500/8 = 62 remainder of 4 2) 600/8 = 75 remainder of 0 3) 700/8 = 87 remainder of 4 4) 4x4 = 16 5) 16/8 = 2 no remainder

Formula for division Example: when a certain number X is divided by 82, the remainder is 5. What is the remainder when X+11 is divided by 41?

-x/y = Q + r/y -ALTERNATE, if stuck on this, try more practical approach to not rely solely on this formula or pick numbers -Remainders must be less than divisor 1) set up equation: X/82 = Q + 5/82 2) isolate X: X = 82Q + 5 3) add 11 to each side: X + 11 = 82Q = 16 4) divide by 41: X + 11/41 = 2Q + 16/41 5) remainder is 16

Formula for Division: Data Sufficiency DS: What is the remainder when positive integer n is divided by 6? 1) When n is divided by 12 the remainder is 9 2) When n is divided by 15 the remainder is 6

1) (n = 12Q + 9)/6 = 2Q + 3, given Q is divisible remainder is always 3 thus sufficient 2) (n = 15Q + 6)/6 = 15Q/6 + 0, given Q is not divisible remainder is not always 0 thus insufficient

Remainder patterns in powers Example: What is remainder when 3^123 is divided by 4?

1) 3/4 remainder of 3 2) 9/4 remainder of 1 3) 27/4 remainder of 3 4) 81/4 remainder of 1 5) pattern is 3,1.... 6) since 123 is off power of 3, remainder of 3

Beads in a necklace Example: Necklace in order of R,B,Y,P. If first bead is R and last is B, all of following can be correct except... 1) 22 2) 44 3) 66

1) 4 beads, plus 2 more, thus pattern is 4n +2 2) 22=4n+2, no remainder thus works 2) 44=4n+2, remainder thus does not work and correct 4) 66=4n+2, no remainder thus works

Remainders for exponents What is remainder when 5^16-3^16 is divided by 16?

1) Quadratic rule: 5^16-3^16 = (5^8-3^8)(5^8+3^8) 2) Continue quadratic rule: (5^8-3^8) = (5^4+3^4)(5^4-3^4) 3) Again and again:....(5-3)(5+3) = 2*8 = 16 4) 16/16 = remainder of 0

Word problems with divisibility Example: Rug was originally priced at W, where W is a whole number. During a liquidation sale, the rug was sold for 6% of its original price. Which of the following could be the sale price of the rug?

1) W = (6/100)*N (N = sale price) 2) (100/6)*W = N 3) (50/3)*W = N 4) Sale price must be multiple of 3

Determining the largest number of PRIME number X that divides into Y! Example 1: What is largest integer value of k, such that 400!/5^k is an integer? Determining the largest number of NON-PRIME number X that divides into Y! Example 2: If 90!/15^n is an integer, what is largest possible value of n? Determining the largest number of NON-PRIME number X=P^k that divides into Y! Example 3: 100!/8^n is integer, what is largest value of n?

Example 1 1) 400/5= 80 2) 400/5^2= 16 3) 400/5^3= 3 4) 80+16+3=99 Example 2 1) 15 = 3*5 (use 5 since higher number) 2) 90/5=18 3)90/5^2=3 4) 18+3=21 Example 3 1) 8 = 2^3, thus divide by 2 2) 100/2+100/4+100/8+100/16+100/32+100/64=97 3) 2^97/2^3n = 3n <=97 4) n<32.33 ~ 32

LCM of X and Y is p and GCF of X and Y is q... Example: LCM of X,Y is 120. GCF of X,Y is 6. What is X x Y?

xy = LCM (x,y) x GCF (x,y) Example: 120 x 6 = 720