Base System

In the base system 10, we use 10 digits. They are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.  There is no 10 in the decimal system.  Similarly, In the base system of 7, we use digits 0 to 6 but 7 won't exist.  To write 7 in base system 7, we use 10.
Converting a decimal system in to any base system:
Suppose for example, We have to convert ${\left( {134} \right)_{10}}$ to base 7.  Then the following process is to be employed.
converting to base 7
So ${\left( {134} \right)_{10}}$ = ${\left( {251} \right)_7}$
We can easily convert a number in any base other than 10 to base system 10.
${\left( {251} \right)_7}$ =  $2 \times {7^2} + 5 \times 7 + 1$ = ${\left( {134} \right)_{10}}$

Find the following table in various base systems:

decimal numbers in different bases

Solved Examples

1. In base 7, a number is written only using the digits 0, 1, 2, .....6. The number 135 in base 7 is 1 x ${{7^2}}$ + 3 x 7 + 5 = 75 in base 10. What is the sum of the base 7 numbers 1234 and 6543 in base 7.
a. 11110
b. 12010
c. 13010
d. 41220

2. If in a certain number system the difference of 5333 and 555 is 4445 then the sum of the numbers 5333 and 555 is
a. 1221
b. 5221
c. 7221
d. 6221

3. 53 x 22 = 1276 then ${(4221)_n} = {({\rm{    )}}_{10}}{\rm{ ?}}$.
a. 1275
b. 3897
c. 4589
d. 5469

4. On planet Jupiter the people use a certain number system to the base ‘n’  (n > 2), Jerk, a resident of the planet, one day received twice his daily wage because the digits of this wage, which was a 2 digit number, were reversed.  If the value of ‘n’ is the least possible value there the decimal representation of the difference between Jerk’s correct wage for the day is
a. 5
b. 8
c. 9
d. 10

5. I take a four-digit number and subtract from it the sum of its digits. In the result I strike off one of the digits and the remaining three digits of the result are 2, 4 and 6 (not necessarily in that order). Find the digit struck off by me
a. 2
b. 4
c. 6
d. 8

6. Once upon a time in ancient times there was a king who was very found of wines. He had a huge cellar, which had 1000 different varieties of wine all in different caskets (1000 caskets in all). In the adjoining kingdom there was a queen who was envious of the king’s huge wine collection. After some time when she could not bear it any more she conspired to kill him by poisoning all his wine caskets. So she sent one sentry to poison all the caskets, but no sooner had the sentry poisoned only one wine casket, he was caught and killed by the Royal guards. Now the king had a major problem in his hand so as to identify the poisonous casket, which he gave to the Minister. But the situation had two peculiarities.
I:      Any one who takes even one drop from the poisonous casket will die.
II:      He will die only after one month.
The king also handed over few prisoners to the Minister as “taster” of those caskets, as their lives was of little value.
If the Minister is allowed only 1 month to find out the poisonous casket, what is the minimum number of prisoners he should use as “tasters”?
a. 10
b. 500
c. 999
d. 1000

7. A three digit non-zero number 'abc' in base 5, when converted to base 7, becomes 'cba'. Which of the following is necessarily true?
a. a must be 2
b. c must be 2
c. b must be 0
d. both a and b