Problem Solving

Each bead in an urn is marked with a distinct positive integer and colored according to that integer's remainder after division by 5, as shown in the following table:

Color Remainder
Red 0
Blue 1
Green 2
Yellow 3
Orange 4

Four blue beads, three green beads, two yellow beads, and one orange bead are withdrawn. If the product of the numbers on these beads is displayed on another bead, according to the rules above, then the color of that bead is...


(A) red
(B) blue
(C) green
(D) yellow
(E) orange

IMO Orange

Red---Nil
Blue--- 4
Green ---3
Yellow------2
Orange----1

so, product= 4x3x2x1=24
24/5===>Remainder 4

Equivalent of 4 is orange from the table given.

I am not sure if I need to consider 0(nil) beads for Red. In case I consider that, the product will be = 0 and hence remainder will be 0, correspondingly the equivalent color will be Red

ankur.agrawal wrote:Each bead in an urn is marked with a distinct positive integer and colored according to that integer's remainder after division by 5, as shown in the following table:

Color Remainder
Red 0
Blue 1
Green 2
Yellow 3
Orange 4

Four blue beads, three green beads, two yellow beads, and one orange bead are withdrawn. If the product of the numbers on these beads is displayed on another bead, according to the rules above, then the color of that bead is...


(A) red
(B) blue
(C) green
(D) yellow
(E) orange

Since the blue beads each have a remainder of 1, the integers on the blue beads are each 1 more than a multiple of 5: 6, 11, 16, 21...
Since the green beads each have a remainder of 2, the integers on the green beads are each 2 more than a multiple of 5: 7, 12, 17, 22...
Since the yellow beads each have a remainder of 3, the integers on the yellow beads are each 3 more than a multiple of 5: 8, 13, 18, 23...
Since the orange beads each have a remainder of 4, the integers on the yellow beads are each 4 more than a multiple of 5: 9, 14, 19, 24...

The lists associated with each color include different units digits:
The blue beads each have a units digit of 1 or 6.
The green beads each have a units digit of 2 or 7.
The yellow beads each have a units digit of 3 or 8
The orange beads each have a units digit of 4 or 9.
To determine the color of the bead in question, we need to know its units digit.

Let Bead X = the bead in question.
Bead X = the product of the integers written on all the other beads.

The units digit of a product = the product of the units digits of each factor.

We can plug in for the units digits of the other beads drawn from the urn:
Product of 4 blue beads = 1*1*1*1 = 1.
Product of 3 green beads = 2*2*2 = 8.
Product of 2 yellow beads = 3*3 = 9.
1 orange bead = 4.

Multiplying the results above, we get:
1*8*9*4 = 288.

Since the units digit of the product above is 8, Bead X must be yellow, the only color that includes a units digit of 8.

The correct answer is D.

OA D

Thanks Guru. Ur explanation cleared my doubt.

ankur.agrawal wrote:Each bead in an urn is marked with a distinct positive integer and colored according to that integer's remainder after division by 5, as shown in the following table:

Color Remainder
Red 0
Blue 1
Green 2
Yellow 3
Orange 4

Four blue beads, three green beads, two yellow beads, and one orange bead are withdrawn. If the product of the numbers on these beads is displayed on another bead, according to the rules above, then the color of that bead is...


(A) red
(B) blue
(C) green
(D) yellow
(E) orange

I guess you got the answer already, however here is how I tried (by the way, it is similar to GMATGuruNy's approach):

We need to find the unit digit of :
(5k1+1)^4*(5k2+2)^3*(5k3+3)^2*(5k4+4)^1 = 1^4*2^3*3^2*4^1 = 1*8*9*4 = 288

Remainder of 288 when divided by 5 = 3, so the color is : yellow D

I still didnt get it

The no of blue balls - 4 possible units digit - 1, 6
The no of Green balls - 3 possible units digit - 2, 7
The no of yellow balls - 2 possible units digit - 3, 8
The no of orange balls - 1 possible units digit - 4, 9

Why are we assuming that all 4 blues balls will have a units digit of 1 or 6, it can so happen 3 are 1's one is 6 and so many other possibilities

Similarly for other colors as well.

How are those conditions taken into account?

What about a scenario where:

blue has all units digits with a 6
green has all with a 2
yellow has all with 8
orange has all with a 9??

Consider This

Every Blue Ball can be represented as (5B + 1)
Every Green Ball can be represented as (5G + 2)
Every Yellow Ball can be represented as (5Y + 3)
Every Orange Ball can be represented as (5O + 4)

If you multiply all the values of drawn balls you get

(5B+1)^4.(5G+2)^3.(5Y+3)^2.(5O+4)
Now all the terms of this equation will be a multiple of 5 except the term that is obtained by multiplying all the remainders 1^4.2^3.3^2.4^1
For example
(5a+1).(5a+1) = 25a^2 + 10a + 1
(5a+1).(5b+2) = 25ab + 10a + 5b + 2.1


This will be applicable in any combination of numbers matching the remainder requirement

Assumed values 4 Blue balls 11,21,16,26 --> (5x2 +1).(5x4 +1).(5x3 +1).(5x5 +1)
Assumed values 3 Green balls 22,37,87 --> (5x4 +2).(5x7 +2).(5x17 +2)
Assumed values 2 Yellow balls 33,88 --> (5x6 +3).(5x17 +3)
Assumed values 1 Orange Ball 109 --> (5x21 +4)

When we multiply all the individual values we will get an equation as below
(5x2 +1).(5x4 +1).(5x3 +1).(5x5 +1).(5x4 +2).(5x7 +2).(5x17 +2).(5x6 +3).(5x17 +3).(5x21 +4)

Now in the above equation all the terms will be a multiple of 5 except 1^4.2^3.3^2.4^1
Therefore when we solve 1^4.2^3.3^2.4^1 we get a remainder that derives the color of the representive ball.