Nups1324 wrote:
Bunuel wrote:
marcodonzelli wrote:
If x is a positive integer and 10^x – 74 in decimal notation has digits whose sum is 440, what is the value of x?
A. 40
B. 44
C. 45
D. 46
E. 50
10^x is a (x+1)-digit number: 1 followed by x zeros.
10^x - 74 is a x-digit number: x-2 9's and 26 in the end. Thus the sum of the digits is (x-2)*9+2+6=440 --> x = 50.
Answer: E.
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Hi
Bunuel,
I didn't understand this part,
"10^x - 74 is a x-digit number: x-2 9's and 26 in the end. Thus the sum of the digits is (x-2)*9+2+6=440 --> x = 50."
It will be great if you'd help me out with this.
Tagging others just in case Bunuel is not able to reply.
yashikaaggarwal chetan2u GMATinsight IanStewart nick1816 fskilnik.
Thank you
Posted from my mobile device=> So, let's say x value is 2
10^2 = 100
10^2 -74 = 26 .
=> If x = 3
10^3-74 = 926
=> If x = 4
10^4-74 = 9926
You can see the pattern that after every power of 10 except 1&0 (because 10^1 = 10 & 10^0 = 1 would have made result in negative) result in same unit and ten digits.
We need to have the power of x, which gives number equivalent to the sum of 440.
In X = 2
10^2-74 = 26 , left digits sum = 2+6 = 8
In X = 3
10^3-74 = 926 , left digits sum = 9+2+6 = 17 and so on....
So we know out of 440 , 8 is the sum of ten and unit digit
440-8 = 432
When we divide 432 by 9 (because 10^x-74 will leave digits as 9999....................99926, we have to find the number of 9)
432/9 = 48
So the number of 9 before 26 is 48 (we are seeking value of x which leave a 50 digit value when 74 is deducted)
If you see the repeated pattern above 10^2-74 left 2 digit value
10^3 left 3 digit value
So 10^50 will leave 50 digit value where there are 48 9's till hundredth value 2 as ten and 6 as one's.
Hence answer is E
Wow.! That is so beautifully explained. This is a really nice concept. I honestly learned something new today.