Question

If x is a positive number such that 2x+12 is perfectly divisible by x, then the number of possible values of x is

Answer 1

so you are given that (2x+12)/x is an integer

Answer 2

my first idea is let (2x+12)/x=k
where k is that integer

Answer 3

play with this equation

Answer 4

big hint: group all your x terms on one side

Answer 5

oops plus get rid of the division part :p

Answer 6

I got it

Answer 7

answer is 6

Answer 8

another question

Answer 9

that's right

Answer 10

Let K be a positive integer such that k+4 is divisible by7. Then the smallest positive integer n, greater than2, such that k+2n is divisible by 7 equals?

Answer 11

k + 4 is divisibly by 7 implies there exists an integer 'a' such that

k + 4 = 7*a

Answer 12

ok

Answer 13

k + 2n is divisibly by 7 implies there exists an integer 'b' such that

k + 2n = 7*b

Answer 14

solve 'k' from first equation and plug it in the second equation

Answer 15

oh

Answer 16

the 2 is given for creating a confusion?

Answer 17

We're not done yet

Answer 18

oh

Answer 19

Have you plugged the expression for 'k' in the second equation ?

Answer 20

i get a=b

Answer 21

7a-4+4=7b

Answer 22

How has 2n turned into 4 ?

Answer 23

oops,wait

Answer 24

7a-4+2n=7b

Answer 25

You have so far :
k + 4 = 7*a
k + 2n = 7*b

solving 'k' from first equation and plugging it in second equation gives
7a - 4 + 2n = 7b

Answer 26

Yes, rearrange a bit and get

-4 + 2n = 7(b-a)

Answer 27

Since right hand side is divisible by 7,
we must choose "n" such that the left hand side is also divisible by 7

Answer 28

For what values of "n" is the expression "-4 + 2n" divisible by 7 ?

Answer 29

hmmm

Answer 30

thinkin, wait

Answer 31

Plugin values like n = 0,1,2,3, etc and see which one makes the expression -4 + 2n divisible by 7

Answer 32

9

Answer 33

good, how about n=2

Answer 34

it becomes zero

Answer 35

Is 0 divisible by 7 ?

Answer 36

7*0=0, so ya

Answer 37

Nice, notice that 9 = 2+7

Answer 38

Can you guess another value for n that makes the expression -4 + 2n divisible by 7 ?

Answer 39

yes

Answer 40

do it

Answer 41

16

Answer 42

how ?

Answer 43

i will tell you,

Answer 44

-4+2n=28

Answer 45

n=16

Answer 46

21 is not possible, so i took 28

Answer 47

Notice again

16 = 2 + 7*2

Answer 48

See the pattern ?

Answer 49

Any integer of form " 2 + 7*t " makes the expression "-4 + 2n" divisible by 7

Answer 50

For example,
n = 2 + 7*3 works just fine

Answer 51

ok, cool

Answer 52

thanks.
Your motivation encourages me to solve more problems.
If I get problems I will let you know :)

Answer 53

Np :) See if you can figure out why any integre of form "2 + 7*t" makes the expression "-4+2n" divisible by 7

Answer 54

when you're free

Answer 55

honestly I did by counting and checking if it was divisible by 7

Answer 56

putting the values 0,1,2,3,4....

Answer 57

that pattern.. man, you are a genius.
How did you take out that!

Answer 58

I think you just put it in format 2+7t after placing the n=0,1,2,3,4, so on

Answer 59

3,4,5,6,8,14
Refer to the Mathematica attachment.