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# [Smart Math] Arithmetic Problem 25

Here’s and example of a SMART MATH problem for ARITHMETIC.

### Problem

What is

$(3333)_{10}$

written in the base system in which $(6822)_{10}$ is written as 5142?

1. 2560
2. 4320
3. 1220
4. 7986
5. 3454

### The Usual Method

First find the number system in which 5142 is equivalent to 6822 in the decimal system (system with base 10). Let ‘x’ be the number system in which 5142 is written. Hence,

$5(x)^{3}+1(x)^{2}+4(x)^{1}+2(x)^{0}$ = 6822

$\therefore 5x^{3}+x^{2}+4x+2=6822$

$\therefore 5x^{3}+x^{2}+4x=6820$

Using remainder theorem of polynomial division, you will find that x = 11

Hence base system is 11.

Now convert

$(3333)_{10}$

to the base system of 11.

This can be done by the division method as follows:

\begin{align} & 11\left| \!{\nderline {\, 3333 \,}} \right. -0 \\ & 11\left| \!{\nderline {\, 303 \,}} \right. -6 \\ & 11\left| \!{\nderline {\, 27 \,}} \right. -5 \\ & 11\left| \!{\nderline {\, 2 \,}} \right. -2 \\ \end{align}

The numbers on the right are the remainde\$ Reading the remainders bottom up, we get the number as 2560. Thus

$(3333)_{10}$

= $(2560)_{11}$.

(Ans: 1)

Estimated Time to arrive at the answer = 150 seconds.

### Using Technique

Observe that 6822 > 5142 per se. Hence number system of 5142 > that of 6822 i.e. 10.

Assume number system as 11 and check as follows:

$5(11)^{3}+1(11)^{2}+4(11)^{1}+2(11)^{0}=6822$. This satisfies the equation as hence our assumption of 11 as the base number system is correct.

Now the number 3333 has to be written in the number system of 11.

Note that the last digit of the answer will be ‘0’ as 3333 is completely divisible by 11 and so the remainder will be 0. (Note that to convert the a number in the decimal system to some other system, we have to divide the number and read the remainders bottom up as is done in the long explanation above.)

This leaves us with options ‘1’, ‘2’ and ‘3’ only. Amongst these options, option ‘2’ is eliminated since 4320 > 3333. As the base of a number system increases, the number per se decreases. However, in this option, $(4320)_{11}>(3333)_{10}$ which is incorrect and so option ‘2’ is ruled out.

Option ‘3’ is also eliminated since 1220 << 3333. Although, as explained in the above paragraph, as the base of a number system increases, the number per se decreases, but there is a certain relationship between the ‘size’ of a number and the base of the number system. Since in this case, the base system changes from 10 to 11, which is a relatively marginal increase, the decrease in the ‘size’ of the number should also be relatively small. However, in case of option ‘3’, the number $(1220)_{11}$ is far smaller than

$(3333)_{10}$

and so it is eliminated.

This leaves us with option ‘1’ which is the answer.

(Ans: 1)

Estimated Time to arrive at the answer = 45 seconds.
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