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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

=> $(xy)_{11}=\frac{1}{3}(yx)_{19}$
=> $y+11x=\frac{1}{3}(x+19y)$
=> $3y+33x=x+19y$
=> $32x=16y$ or $2x=y$