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# CAT Questions On Base System PDF:

Download CAT Questions On Base System PDF. Practice important problems with detailed explanations for CAT on Base system related to conversions and sums.

Question 1: What will be the number of zeroes in$(2000!)_{34}$. Here 34 is the base in which the number is written.

a) 122
b) 123
c) 124
d) 125

Question 2: $(245)_{x}+(162)_{x}-(427)_{x}=0$ in some base x. What is the value of x?

a) 8
b) 9
c) 10
d) Cannot be determined

Question 3: A number N has 4 digits when expressed in base 8, 3 digits when expressed in base 11 and 3 digits when expressed in base 26. How many values can N take?

a) 164
b) 298
c) 474
d) 655

Question 4: How many 3 digit numbers in decimal system are 3 digit numbers in both base 9 and base 11?

a) 625
b) 507
c) 528
d) 608

Question 5: A two digit number A in base 11 is one-third of the number formed by reversing its digits when considered in base 19. How many such numbers are possible?

a) 2
b) 1
c) 5
d) 4

34 = 17*2
So we have to find the highest power of 17 in 2000!. We need not find the power of 2 because of 2 will be greater than the power of 17. Thus the power of 17 will act as the limiting value.
Thus the highest power of 17 in 2000! is
[2000/17] + [ 2000/289] + [2000/4913], [] is greatest integer function
= 117 + 6 + 0 = 123
Thus the required number of zeroes is 123.

$(245)_{x}+(162)_{x}-(427)_{x}=0$
$=>(2x^2+4x+5)+(x^2+6x+2)-(4x^2+2x+7)=0$
$=>x^2-8x=0$
$=>x=8$

$8^3 = 512$ and $8^4 = 4096$ => The minimum value and maximum value of a number in base 10 to have 4 digits in base 8 are 512 and 4095 respectively => range = (512,4095).
$11^2 = 121$ and $11^3 = 1331$ => The minimum value and maximum value of a number in base 10 to have 3 digits in base 11 are 121 and 1330 respectively => range = (121,1330).
$26^2 = 676$ and $26^3 = 17576$ => The minimum value and maximum value of a number in base 10 to have 3 digits in base 26 are 676 and 17575 respectively => range = (676,17575).
Common range = (676, 1330)
=> Number of values that N can take = 1330 – 676 + 1 = 655

Smallest 3 digit numbers in base 9 is $1*9^{2} + 0* 9 + 0* 9^{0} = 81$, Largest 3 digit number in base 9 is $8*9^{2} + 8*9 + 8*9 = 728$
Smallest 3 digit number in base 11 is is $1*11^{2} + 0* 11 + 0* 11^{0} = 121$

Largest 3 digit number in base 11 is $10*11^{2} + 10*11 + 10*11^{0}= 1330$

So 3 digit numbers in base 10 which are also 3 digit numbers in base 9 and base 11 are
from 121 to 728 = 608

Let the number be xy.
=> $(xy)_{11}=\frac{1}{3}(yx)_{19}$
=> $y+11x=\frac{1}{3}(x+19y)$
=> $3y+33x=x+19y$
=> $32x=16y$ or $2x=y$
So, (x,y) can be (2,1)(4,2)(6,3)(8,4)(10,5)
Numbers such as 10, 11, 12.. etc. are represented as alphabets a,b,c etc in higher bases. Thus, the case of (10,5) also satisfies our requirements.
The next pair (12,6) can’t be considered since 12 is not a single digit number in base 11 and thus, all other cases greater than (10,5) can be excluded.