how to solve this one?

[themostimpthing]

The function f is defined for each positive three-digit integer n by f(n) = 2x3y5z , where x, y and z are the hundreds, tens, and units digits of n, respectively. If m and v are three-digit positive integers such that f(m)=9f(v), them m-v=?
(A)8
(B)9
(C)18
(D)20
(E)80
Please help..

[Test Taker]

I'm bit confused with the Q.

Did u mean (2^x)*(3^y)*(5^Z)?

^ means power.

If so my ans would be D.

[RamN]

The approach will be to back calculate...

consider n = 111. F(n) will be 30.. 9 * F(n) = 270

Now break 270 into its factors and rearrange so tht it matches F(m) = 2 * 27 * 5... this implies x =1, y = 9, z = 1...

m - n = 191 - 111 = 80

[suwarnapatel]

Simpler method:

f(m)=9f(v)
Try 9 against 2,3 and 5

3^2 = 9

So we know tens digit is greater by 2 in m than v.

difference = 10 (tens digit) * 2 = 20.

[nerrad_oh]

There seems to be differing opinion here. Which is the correct way? Is the correct answer = 20 or 80 ??

[ssky]

Lets look at the question in pieces

Number is n comprising of xyz where x is 100th, y is 10 and z is unit digit

function f is defined as f(n) = 2x3y5z

Eg. if n=112 then F(n) = 2(1)*3(1)*5(2)=60

Now if f(m)=9f(v) then what? I am lost after that
I know there must be some way of finding the m and v values isntead of guessing.

[Queen09]

Do you mean (2^x)*(3^y)*(5^Z) ???

[nerrad_oh]

IMO answer is A = 8 ??

Given f(n) = 2x3y5z. So if n = 123, this means 2(1)*3(2)*5(3)= 180.

Given f(m) = 9*f(v). Just assume 3-digit number m = abc and v = def.

f(m) = 9*f(v)
2a3b5c = 9*[2d3e5f]
(2*3*5)abc = 9*(2*3*5)def
30abc=9*30def
abc=9*def

Now m-v = abc-def = 9*def-def = 8.

But I look at RamN answer and that seems to make sense too
But doesn't that answer assumes n=111? If n=222, will the same holds true ?

[shekhar000]

IMO u cannot get an answer here if we assume 2x3y5z mean 2*x*3*y*5*z
if f(m) = 9(v), if we assume that xyz for m and abc for v, xyz = 9 abc.
Now, x , y, z will individually have a relation with a b c which will be based on 3 factors of 9, i.e. 1 *9*1 and 1*3*3.
with each set, if v is abc we can have M as = [(a*1)(b*9)(c*1)] or (a*9)(b*1)(c*1) or (a*1)(b*1)(c*9)
this will give you m -v as 80, 800, or 8.
if we have the other set, M will be [(a*1)(b*3)(c*3)] or (a*3)(b*3)(c*1) or (a*3)(b*1)(c*3)
then m-v can be any value. as we have two possible digits if multiplied by 3 gives us another digit. so if u assume v to 222
then m is 266, 662, 626 / and if v is 333 then it m can be 399, 993, 939 so m-v can take different values.

[amit_ace]

Based on the given question I guess we can have two answers
(A)8
(B)9
(C)18
(D)20
(E)80
for the F(m)=9F(V) we need to go by given answers.
either we should have product of two digits of F(m) having 9 as factor.
Only possible corresponding 'different' digits are V(M) 1(3),2(6),3(9) or 1(9) X,y,Z <> 0

Out of given choices
8 means difference only in units place
8 is possible 111 ,119 or 121 ,129 -there can be numerous examples -This is correct

9 means diff only in units place (or difference of factor of 3 each in units and tenth place which is not possible as tens digit difference would be such that total diff more than 9)
9 is not possible. Since it will leave one of the numbers zero which will make f(v)=0

18 means diff in units and tenth place.Now we cant have a difference of 9 in units or tenths place alone (as it would make function zero for either m or n)
therefore each unit and tenth place of m should have a factor of 3 more than
that of v .Only possible corresponding digits are V(M) 1(3),2(6),3(9)
clearly using these difference of 18 is not possible .Minimum diff is 22
X 33 X 11, other combination has a bigger difference eg X 3 6 X 12

Again for 20 it is clear that difference is not in units place.
It is tens place or
you can say difference of twenty in 100s and 10s place together eg 21Z and 19Z.- this is not possible as
Now we cannot have two digits with a difference of 2 if one digit is 9 times the other one

80- this is possible 111 and 191 so this is also correct

Please let me know if you have alternate views