Quant Boosters  Vikas Saini  Set 2

Let's start from base 3.
smallest 5 digit number in base 3 = (10000)_3 = 3^4.
Largest 5 digit number in base 3 = (22222)_3 = 2(11111)_3 = 3^5  1.
in base 3 not possible.
Now let's take base 4.
smallest number = (10000)_4 = 4^4 = 256.
Largest number = (33333)_4 = 4^5  1 = 1023.
256 < 677 < 1023.
Hence base 4.

Q26) How many 3 digit numbers are there in base 9 system.

Hundred place = 1 to 8 = 8 digit.
Ten's place = 0 to 8 = 9 digit.
Unit's place = 0 to 8 = 9 digit.
Total = 8 x 9 x 9 = 648 digits.

Q27) How many base 10 numbers are three digit numbers in base 7 and four digit numbers in base 6.

Smallest 3 digit number in base 7 = (100)_7 = 7^2 = (49)_10.
Largest 3 digit number in base 7 = (666)_7 = 7^3  1 = (342)_10.
[49,342] numbers are 3 digits numbers in base 7.
Smallest 4 digit number in base 6 = 6^3 = 216.
Largest 4 digit number in base 6 = 6^4  1 = 1295.
[216,1295] numbers are 4 digit numbers in base 6.
[216,342] common numbers.
Total decimal numbers = 342  216 + 1 = 127.

Q28) N in base 10 = aaa in base 3. Here a is a single digit number, then N is a multiple of.
(a) 9
(b) 3a
(c) 13
(d) 26.

(aaa)_3 = a (111)_3
a (3^2 + 3 + 1)
(13a)_10 = (N)_10.
Hence 13.

Q29) How many 3 digits are there in decimal system which have exactly 3 digits when expressed in both base 7 & base 13.

Smallest 3 digit number in base 7 = (100)_7 = 7^2 = 49.
Largest 3 digit number in base 7 = (666)_7 = 7^3  1 = 342.
[100,342] are the number which are exactly 3 digit number in base 7.
Smallest 3 digit number in base 13 = (100)_13 = 13^2 = 169.
Largest 3 digit numbers in base 13 = (CCC)_13 = 12^3  1 = 1727.
[169,999] are the three digit numbers which are exactly 3 digit numbers in base 13.
[ 169,342] is common in both.
Total = 342  169 + 1 = 174 numbers.

Q30) In base x, 206 + 216 = 424. What is value of x ?

2x^2 + 6 + 2x^2 + x + 6 = 4x^2 + 2x + 4
= > x + 12 = 2x + 4.
= > x = 8.