A sequence of 4 digits, when considered as a number in base 10 is four times the number it represents in base 6. What is the sum of the digits of the sequence?(CAT, 2016)
- May 15, 2018
- Posted by: allexamshelps
- Category: CAT
4,956 total views, 1 views today
A sequence of 4 digits, when considered as a number in base 10 is four times the :
In base n, the four digits are n0= 1, n1= n, n2= 2n, n3= 3n. That is if you want to write 4 digits in base n, then exponents of n will start from 0 to 3.
Let's understand it:
Further, the question says that there are four digits in a sequence; let’s say the four unknown digits are: pqrs
Following the above rule, the four digits in base 6 are:
And, in base 10 are:
It is given that 10base is 4 times of 6base:
10base is 4 x 6base or 10base = 4 (6base); or
4 x (6base) = 10base
(a=b and b=a are equal)
4 x (216p+36q+6r+s) = 1000p+100q+10r+s
(Open the bracket multiplying each term into the bracket by 4 which is the common multiplier.
4 x (216p+36q+6r+s) = 864p+144q+24r+4s
Arrange like terms together
(864p-1000p)+(144q-100q)+(24r-10r)+(4s-s) = 0
44q+14r+3s=136p now we have to choose values for p, q, r, and s arbitrarily, so that this equation may hold true.
By trial and error approach, we have found that p=1, q=2, r=3, and s=2
( Actually what we did that we tried to equate 44q+14xr+3s to 136p, and we had put p=1 before. We then found that q=2; r=3 and s=2 can make L.H.S= R.H.S (136)
44 x 2+14 x 3+3 x 2=136 x 1
This means p=1; q=2; r=3 and s=2
When we add these digits:
Hence the option (d) is correct.
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