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Each bead in an urn is marked with...

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Each bead in an urn is marked with...

by BTGmoderatorLU » Fri Oct 27, 2017 6:14 am
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:

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

15 blue beads, 18 green beads, 13 yellow beads, and 3 orange bead are withdrawn.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

The OA is D.

I need help to solve this PS question. Can any expert explain it for me please? Thanks.
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Answer

by EconomistGMATTutor » Sat Oct 28, 2017 3:08 am
Hello LUANDATO. I hope you're well.

This is a tough question, but let's take a look at it.

We only have to take care about the remainders.

* It has been removed 15 numbers whose remainder when divided by 5 is 1.

The product of these 15 numbers could be write as $$R_1=\left(5n_1+1\right)\cdot\cdot\cdot\left(5n_{15}+1\right).$$

* It has been removed 18 numbers whose remainder when divided by 5 is 2.

The product of these 18 numbers could be write as $$R_2=\left(5m_1+2\right)\cdot\cdot\cdot\left(5m_{18}+2\right).$$

* It has been removed 13 numbers whose remainder when divided by 5 is 3.

The product of these 13 numbers could be write as $$R_3=\left(5j_1+3\right)\cdot\cdot\cdot\left(5j_{13}+3\right).$$

* It has been removed 3 numbers whose remainder when divided by 5 is 4.

The product of these 3 numbers could be write as $$R_4=\left(5h_1+4\right)\left(5h_2+4\right)\left(5h_3+4\right).$$

Now, we can see that $$\left(5t_1+r\right)\cdot\cdot\cdot\left(5t_P+r\right)=5W+r^P.$$ So,

$$R_1=5W_1+1^{15}=5W_1+1.$$
$$R_2=5W_2+2^{18}=5W_2+262144=5W_2+262140+4$$ $$=5W_2+5\cdot52428+4=5\left(W_2+52428\right)+4.$$ $$R_3=5W_3+3^{13}=5W_3+1594323=5W_3+1594320+3$$ $$=5W_3+5\cdot318864+3=5\left(W_3+318864\right)+3.$$
$$R_4=5W_4+4^3=5W_4+64=5W_4+60+4$$ $$=5W_4+5\cdot12+4=5\left(W_4+12\right)+4.$$

Finally, the product of ALL numbers is $$T=R_1\cdot R_2\cdot R_3\cdot R_4=5H+1\cdot4\cdot3\cdot4=5H+48=5H+45+3$$ $$=5H+5\cdot9+3=5\left(H+9\right)+3.$$

It implies that the remainder when T is divided by 5 is equal to 3. So, the color of the bead should be YELLOW.

So, the correct answer is D.

I really hope this can help you.

As I said, we are only interested in the remainders.

Feel free to ask me any doubt you could have.
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