Question

S.O.S: I need help with this: prove that if a+1 divides b and b divides b+3, then a=2 and b=3. Proofs I have covered are direct, contrapositive, and contradiction!!

Answer 1

if a+1 divides b, this means that b = k *(a+1) for some integer k.
if b divides b+3 this means that b+3 = mb for some integer. ok so far?

Answer 2

right that is what I have so far

Answer 3

ok lets subtract the two equations

Answer 4

(b+3) - b = mb - (a (k+1))

Answer 5

so subtract b+3-b?

Answer 6

yes

Answer 7

yes

Answer 8

system of equations

Answer 9

i get 3 = mb - ak - a

Answer 10

that is what I got too

Answer 11

is there more info
are a and b must be integers?
well yes if we are discussing "divides"

Answer 12

ok now substitute back

Answer 13

suppose a and b are positive integers

Answer 14

nevermind, thats circular to substitute back

Answer 15

wait I have a question

Answer 16

isn't b= (a+1)k? and b+3= bj?

Answer 17

i have a different idea

Answer 18

yes , thats correct

Answer 19

add 3 to the first equation , so b+3 = 3 +(a+1)k

Answer 20

so bj = 3 + (a+1)k

Answer 21

so bj - 3 = (a+1)k

Answer 22

why do you have (b+3)-b= mb-(a(k+1))?

Answer 23

wait you did that backwards

Answer 24

isn't it suppose to be mb-(a+1)k?

Answer 25

b= (a+1)k? and b+3= bj? this is false

Answer 26

if a+1 divides b then b=(a+1)k?

Answer 27

isnt the definition of divisiblity if a divides b then b= ak?

Answer 28

ok so we have bj-3 = (a +1) k ,

Answer 29

yes i misread

Answer 30

how did you get bj-3= (a+1)k?

Answer 31

a few steps

Answer 32

ok first step, b = (a+1)k , b+3 = bj

Answer 33

did you do that by substituting?
i see what you did

Answer 34

3 = bj - (a+1)k

Answer 35

rearranging we get

Answer 36

(a+1)k = bj - 3

Answer 37

but we know that b = (a+1) k

Answer 38

so b = bj - 3

Answer 39

follow so far, i might have gone too fast

Answer 40

b = (a+1)k , b+3 = bj , subtracting the former from the latter we get
3 = bj - (a+1)k

Answer 41

add (a+1)k to both sides, subtract 3 from both sides
you should get bj = (a+1)k ,

Answer 42

I got it so far

Answer 43

woops

Answer 44

bj - 3 = (a+1)k

Answer 45

ok we know that (a+1)k = b, so by transitive rule we have
b = bj - 3

Answer 46

so b+3 = bj , so b divides b+3, which we already know, shoot

Answer 47

ok lets do some substitution

Answer 48

b = (a+1)k , b+3 = bj ,
thats the given.
substitute (a+1)k for b in the second equation

Answer 49

so we get (a+1)k + 3 = (a+1)k*j

Answer 50

rewriting it as b= bj-3 does that mean that we proved b=3 since that can be rewritten as b-3=0?

Answer 51

how did you get that? i got b + 3 = bj

Answer 52

ok i have the answer

Answer 53

bj-3= (a+1) k and we know that b=(a+1)k

Answer 54

so b = bj - 3

Answer 55

which is b + 3 = bj , and we already know this

Answer 56

ok then, ready for the solution

Answer 57

yeah but I am not sure if that proves that b=3?

Answer 58

it doesnt add any new information
its not helpful

Answer 59

so we abandon that

Answer 60

different approach

Answer 61

b = (a+1)k , b+3 = bj ,
thats the given.
substitute (a+1)k for b in the second equation
so we get (a+1)k + 3 = (a+1)k*j

Answer 62

right

Answer 63

good
now divide both sides by a+1

Answer 64

just to make sure we are starting all over?

Answer 65

yes

Answer 66

did you get a+1)k + 3 = (a+1)k*j

Answer 67

i got k+ 3/(a+1)= kj

Answer 68

so 3/(a+1) = kj - k

Answer 69

so 3/(a+1) is an integer

Answer 70

because kj and k are integers, and the difference of integers is always an integer

Answer 71

or 3/(a+1) = k ( j-1)

Answer 72

there is only one positive value that will make 3/(a+1) an integer

Answer 73

is that wrong? I did it twice and that is what i got

Answer 74

a=2 will make 3/(a+1) = 3/3 = 1

Answer 75

or backtrack

Answer 76

we assumed, k, j, and a are positive initially

Answer 77

we can prove this actually

Answer 78

, k, j, a, b are all positive

Answer 79

ok so far?

Answer 80

we'll go back to show that k and j are positive, but lets just assume it is at the moment

Answer 81

so we have k + 3/ (a+1) = kj, agreed?

Answer 82

right but what does the k(j-1) mean in this context?

Answer 83

well we dont need that. that just shows the left left is an integer

Answer 84

but i want to make this airtight

Answer 85

to avoid the negative cases

Answer 86

k + 3/(a+1) = kj , agreed?

Answer 87

dont subtract k from both sides

Answer 88

right hand side is positive, and left hand side is positive

Answer 89

Yes. so we would have 2 cases where k and j are even and then another where k and j are odd

Answer 90

dont need that

Answer 91

for this to be an equality, then we need kj to be a positive integer

Answer 92

err, i mean the left side to be a positive integer, since the right side is already a positive integer

Answer 93

why?

Answer 94

for this to be an equality we need the left side to be a positive integer , because the right side we know is a positive integer , (k , j are both positive integers so the product of two positive integers is another positive integer)

Answer 95

the right side is a positive integer, because k is positive, and j are positive. and we know pos. integer* pos. integer = pos. integer ,

Answer 96

its an axiom, positive integers are closed under multiplication (or a theorem if youre doing peano math, etc)

Answer 97

so then having k+ 3/(a+1)=kj and rewriting it by subtracting k to the other side : 3/(a+1) = k(j-1) wouldn't that alright since k is a positive and j is a positive integer and subtracting 1 from a positive integer is still positive so the right side would be a positive integer

Answer 98

no, it isnt always positive