Integer Properties Flashcards
Divisibility rule for 4
if the last two digits form a two-digit nb divisible by 4, then the entire number is divisible by 4
Divisibility rule for 9
if the sum of the digits is divisible by 9, then the number is divisible by 9
Divisibility rule for 6
in order to be divisible by 6, a number must be divisible by 2 AND divisible by 3
is 1296 divisible by 6?
yes
true or false
divisor = factor
true
7 is a factor of 91
7 is a divisor of 91
91 is divisible by 7
91 is a multiple of 7
If P is a multiple of r, r is a factor of P and a divisor of P
If P and Q are multiples of r, then P+Q and P-Q must also be multiples of r
If P is a multiple of r, then any multiple of P is also a multiple of R
If P and Q are multiples of r, then P*Q must also be a multiple of r
If K, (K+200), (K+350) and 15*K are all multiples of P, then P could equal which of the following? A - 20 B - 25 C - 75 D - 100 E - 150
15*K is a useless piece of information: if K is a multiple of P, then 15K has to be a multiple of P.
differences amount K, K+200 and K+350 will also be multiples of P K + 200 - K = 200 K + 350 - K = 350 350 - 200 = 150 200 - 150 = 50 P is a factor of 50, 150, 200 and 350
The answer can’t be anything bigger than 50 because anything bigger than 50 can’t be a factor of 50
can’t be 20 (others not divisible by it)
All the numbers are divisible by 25
Answer B
List the prime numbers less than 20
2 3 5 7 11 13 17 19
List the prime numbers between 20 and 60
23 29 31 37 41 43 47 53 59
how to test whether a large number (less than 100) is prime?
check whether it is divisible by one of the prime nb less than 10 (2 - 3 - 5 - 7 )
If 4680 = 2^3 * 3^2 * 5 * 13 Which of the following nb are factors of 4680? 25 45 65 85 120 180
25 = 5*5 NO 45 = 3*3*5 YES 65 = 5*13 YES 85 = 5*17 NO 120 = 2^3 * 3 * 5 YES 180 = 2^2 * 3^2 * 5 YES
How to find the nb of factors of an integer?
prime factorization
list of exponents of the prime factors
add one to every nb on the list
multiply all those nb together
How to find the nb of odd factors of an integer?
prime factorization
list of exponents of odd prime factors
add one to each
product of nb on new list = nb of odd factors
How to find the nb of even factors of an integer?
no direct way
calculate the total nb of factors and the nb of odd factors, then subtract
How many factors does 8400 have?
Prime factorization
8400 = 84 x 100 = 7 x 12 x 10 x 10
8400 = 7 x (2 x 2 x 3) x ( 2 x 5 ) x (2 x 5 )
8400 = 2^4 x 3 x 5^2 x 7
make a list of the exponents of the prime factors
{4, 1, 2, 1}
add one to every number on the list
{5, 2, 3, 2}
multiply all those numbers together
5 x 2 x 3 x 2 = 60
8400 has 60 factors
How many even factors does 21600 have?
21 600 = 2^5 x 3^3 x 5^2
{5+1, 3+1, 2+1}
6x4x3 = 72 factors
{3+1, 2+1}
4x3 = 12 odd factors
72-12 = 60 even factors
List the first fifteen perfect squares
1² = 1 2² = 4 3² = 9 4² = 16 5² = 25 6² = 36 7² = 49 8² = 64 9² = 81 10² = 100 11² = 121 12² = 144 13² = 169 14² = 196 15² = 225
The exponents of the prime factors of a square …
… all must be even
If K = 2^6 x 3^4 x 5^2
Is K a perfect square?
Yes
all the exponents of the prime factors are even
K = 360²
A perfect square always has …
… an odd nb of factors
What is the GCF of 720 and 1200?
GCF = Greatest common factor / greatest common divisor
720 = 2^4 x 3^2 x 5 1200 = 2^4 x 3 x 5^2 Highest power of 2 in common : 4 Highest power of 3 in common : 1 Highest power of 5 in common : 1 GCF = 2^4 x 3 x 5 = 240
What is the LCM of 24 and 32?
LCM = Least common multiple = LCD = least common denominator
24 = 2^3 x 3 32 = 2^5 GCF = 8
write each nb in the form : GCF x another factor
24 = 8 x 3
32 = 8 x 4
LCM = product of these three factors LCM = 8 x 3 x 4 = 96
1/10 - 1/35 = ?
GCF of 10 and 35 is 5 10 = 5 x 2 35 = 5 x 7 LCD = LCM = 5 x 2 x 7 = 70 1/10 - 1/35 = 7/70 - 2/70 = 5/70 = 1/14
LCM / GCF formula
LCM = ( P x Q ) / GCF
0 is… (x3)
an even integer
not positive
not negative
Evens and Odds (adding & subtracting)
E + E =
E - E =
O + O =
O - O =
E + O =
E - O =
E + E = E
E - E = E
O + O = E
O - O = E
E + O = O
E - O = O
Add or subtract “likes”, we get EVEN
Add or subtract “unlikes”, we get ODD
Evens and Odds (multiplying)
E x E =
O x O =
E x O =
E x E = E
O x O = O
E x O = E
as long as there is at least one even factor, the product will be even
P, Q, R and S are integers. If P is even and (PxQ + RxS) is an odd integer, then which of the following must be true?
I - Q is odd
II - R is odd
III - S is odd
A : I only B : I and II only C : I and III only D : II and III only E ; I, II and II
D : II and III only
P is an odd integer and (P² + QxR) is an even integer, then which of the following must be true?
A : either Q or R is an odd integer B : either Q or R is an even integer C : both Q and R are odd integers D : both Q and R are even integers E : nothing can be concluded
P² is odd so QxR must be odd
we have no guarantee that Q and R are integers !!!
E : nothing can be concluded
If P and Q are integers, and if (P² + PQ) is an odd number, what do we know about P and Q?
P is odd and Q is even.
either plug 1 and 2 for odd and even nb and try the 4 cases.
or use logic: if P is even, it makes both terms even which doesn’t work (the result would be even) so P must be odd
if P is odd, the first term is odd. If the sum is odd, the second term must be even. if P is odd, the only way PQ can be even is if Q is even
If P and Q are integers, and (4P + Q) is odd, what must be true?
Q is odd
4P must be even, so Q is odd (or use the 4 cases method)
When positive integer N is divided by positive integer P, the quotient is 18, with a remainder of 7. When N is divided by (P + 2), the quotient is 15 and the remainder is 1. What is the value of N?
N/P = 18 + 7/P N/(P+2) = 15 + 1/(P+2) SO !!! N = 18P + 7 N = 15(P+2) + 1 = 15P + 31
18P + 7 = 15P + 31
3P = 24
P = 8
N = 15 (8+2) + 1 = 151
