Question

Fun question...
Three people are sitting at a restaurant table, a grandfather, his son and his grandson.The waiter asked the boss what are their ages, the boss replied "The product of their ages is 4032, the sum of their ages is twice my age and if you divide the grandfather's age by the grandson's age you get twice your age". What is the age of the son?
All ages are integer values

Answer 1

what is the boss's age and the waiter's age?

Answer 2

"the sum of their ages is twice my age (boss) and if you divide the grandfather's age by the grandson's age you get twice your age (waiter)"

Answer 3

3 equations, 5 unknowns....

Answer 4

k : grand son(kid)
s : son
g : grand father
b : boss
w : waiter

ksg = 4032
k+s+g = 2b
g/k = w

Answer 5

solve in naturals

Answer 6

g/k = 2w

Answer 7

i would start here :
http://www.wolframalpha.com/input/?i=prime+factorization+of+4032

Answer 8

k, s, and g could be any three #'s that are factors of 4032 because b and w are not specified.

Answer 9

ohk good catch :) corrected below :

ksg = 4032
k+s+g = 2b
g/k = 2w

Answer 10

let me correct myself, k, s, and g can be any 3 #'s that multiply to be 4032

Answer 11

also one more constrain :
k < s < g

Answer 12

so, the system is :

ksg = 4032
k+s+g = 2b
g/k = 2w
k < s < g

all naturals

Answer 13

nothing funny like the grandfather is the boss or some such thing?

Answer 14

there's more than one answer, since the b and w are not specified.

Answer 15

\(g/k = 2w \implies g = 2kw\)
that gives \(g \) must be even

Answer 16

solve it!! ill give u a hint dont pay too much attention to what age is it really possible to have kids and that stuff lol

Answer 17

the numbers were chosen to just to make it funny i think

Answer 18

just to mess with all those guessers haha

Answer 19

you are on right path!! ganeshie8

Answer 20

the kid cannot have 7 as a factor :

\(k = 2^m3^n\)

\(0\le m \le 2, ~~0\le n \le 1\)

Answer 21

m and n cannot be 0 at the same time,
so that gives 5 cases to consider

Answer 22

oh they can be 0 at the same time as well,
then the kid will be 1 year old

Answer 23

\(k = 1\) seem to fit all constraints ?

Answer 24

the question is actually pretty deep, its a number theory question, but the math isnt complicated its just tricky

Answer 25

u gotta start here
first

Answer 26

xyz=4032
x/z=2W
x+y+z=2B
x=2wz
2wZ^2*Y=4032
Z=sqrt(2016/yw)

Answer 27

try to take it from there

Answer 28

gota do some number theory deductions from there

Answer 29

beautiful !
yw can be : 2*7 or 2^3*7 or 2^5*7 or 2*3^2*7 or 2^3*3^2*7 or 2^5*3^2*7

Answer 30

kid = {12, 6, 3, 4, 2, 1}

Answer 31

it can be any of above 6... thats it right /

Answer 32

theerse more

Answer 33

theres actually a unique answer

Answer 34

i didnt give u much it was just to get started, i dont wanna ruin it for u lol

Answer 35

oh we dint fold in below constraint yet
x+y+z=2B

Answer 36

that constraint actually satisfies all the 6 values for kid acutally

Answer 37

\(\color{red}{\#\# k=1, s = 3^2, g = 2^67}\)

\(ksg = 2^63^27 = 4032\)

\(k+s+g = 2(5 + 2^57)\) = 2b

\(g/k = 2(2^47) = 2w\)



\(\color{red}{\#\# k=2, s = 23^2, g = 2^47}\)

\(ksg = 2^63^27 = 4032\)

\(k+s+g = 2(1 + 3^2 + 2^37)\) = 2b

\(g/k = 2(2^27) = 2w\)

... similarly other 4 cases will also satisfy all 3 given constraints

Answer 38

wat do u mean by an uniq solution ha ?

Answer 39

unique age is not possible for anyone of kid/son/grandpa wid just those 3 constraints right ?

Answer 40

u want solution?

Answer 41

there is a unique solution

Answer 42

wit, clarify my question first

Answer 43

*Wait

Answer 44

ok lemme read

Answer 45

how are above two cases violating the given constraints ?

Answer 46

just read my recent reply where i marked the two lines in "red"

Answer 47

um lemme see there are certain constraints that come up when u see this equation

Answer 48

finding the constraints are actually part of the question

Answer 49

thats most of the hard work

Answer 50

like so u know that
Z=12*sqrt(14/YW)

Answer 51

question boils down to only 3 constraints :
1) product of ages = 4032
2) sum of ages is even number
3) kids age evenly goes in grandpa's age

Answer 52

everything else can be derived based on above 3

Answer 53

lets say that k=P/(somefactor of 12)^2

Answer 54

no more constraints like

Answer 55

im still looking for explanation of how the two cases i gave dont fit the solution

Answer 56

id have to go into some of the deailts of the solution to show u some extra constraints

Answer 57

question boils down to only 3 constraints :
1) product of ages = 4032
2) sum of ages is even number
3) kids age evenly goes in grandpa's age

Answer 58

I thought, those are the only 3 constraints we need to satisfy ?

Answer 59

btw, the sqrt(2016/yw) thing is derived from above.. its not a new constraint :/

Answer 60

i know i was thinking about this too

Answer 61

i was having the same issue where u said there are other possible answers as long as yw is a factor of 12^2 and divides 14

Answer 62

so, the system is :

ksg = 4032
k+s+g = 2b
g/k = 2w
k < s < g

all naturals

Answer 63

can u derive any single other constraint, thats different from the above list /

Answer 64

?

Answer 65

nope,
and the two cases i gave satisfy above constraints.
so the solution is not unique
QED.

Answer 66

^^thats my thinking...

Answer 67

hold on i argued this with my friend too

Answer 68

look at this

Answer 69

dont mind the pellets here and there he likes to swear -.- a lot

Answer 70

thats mouthful lol :/
il go thru this shortly,
but u havent clarified my earlier question yet

Answer 71

so look for p such that 1< r <14 AND (14/r) is a non-prime integer

Answer 72

do u understand this statement here
so look for p such that 1< r <14 AND (14/r) is a non-prime integer

Answer 73

honestly i dont know because like you i tried the other possibilies on the quiz

Answer 74

and it said it was wrong, they were looking for a unique solution and his solution was right

Answer 75

but then agian the question was taken down after a while, or i just couldnt find it again

Answer 76

im not sure if it was taken down because it wasnt a unique solution or i just cant find it -.-

Answer 77

there is a flaw in the solution

Answer 78

where

Answer 79

are u saying it because he said m is a ratio of grandpa to grandson

Answer 80

or "so look for p such that 1< r <14 AND (14/r) is a non-prime integerÈ

Answer 81

"so look for p such that 1< r <14 AND (14/r) is a non-prime integer"

Answer 82

nvm im still going thru it...

Answer 83

here was m solution i didnt fully understand his reasoning for that statement but i got same answer

Answer 84

but this has a flaw because u dont need a perfect square to come outta there
it can be 1/2 1/3 or 1/6 aswell

Answer 85

not 1/4 since then y yould be odd and not 12 since then y<2

Answer 86

then we need (14/mz) = r (because y is an integer)

Answer 87

mz | 12^2*14

not just :

mz | 14

Answer 88

yaa okkay makes sense so i guess the question was taken down because it was flawed

Answer 89

and when it divides, the result must be a perfect square

Answer 90

DANG it wasnt taken down before i lost some rating with a couple guesses

Answer 91

looking for mz | 14 gives unique solution when mz = 14, and y = 12
im not sure still i followed ur solution correctly.. uhmm

Answer 92

however there will be many other solutions when mz | 12^2 * 14 is a perfect square

Answer 93

nevertheless the problem is nice :) we can make the solution unique by changing the problem a bit i hope