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A number is interesting if on adding the sum of the digits of the numb

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Manager
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A number is interesting if on adding the sum of the digits of the numb  [#permalink]

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New post 19 Aug 2019, 10:08
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Difficulty:

  55% (hard)

Question Stats:

45% (03:02) correct 55% (02:09) wrong based on 22 sessions

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A number is interesting if on adding the sum of the digits of the number and the product of the digits of the number, the result is equal to the number. What fraction of numbers between 10 and 100 (both 10 and 100 included) is interesting?

(A) 0.01

(B) 0.1

(C) 0.11

(D) 0.16

(E) 0.23
Director
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Re: A number is interesting if on adding the sum of the digits of the numb  [#permalink]

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New post 19 Aug 2019, 10:59
1
We can write any 2-digit number in the form of 10x+y, where x can take integral values from 1 to 9, and y can take integral values from 0 to 9.

10x+y= x+y+xy
10x=(y+1)x
y+1=10
y=9

Hence, any 2-digit number that has unit digit 9 can be written as the sum of product of its digits, and sum of its digits.
Total possible "interesting" numbers= 9 (19, 29, 39, 49, 59, 69, 79, 89, and 99)


Hovkial wrote:
A number is interesting if on adding the sum of the digits of the number and the product of the digits of the number, the result is equal to the number. What fraction of numbers between 10 and 100 (both 10 and 100 included) is interesting?

(A) 0.01

(B) 0.1

(C) 0.11

(D) 0.16

(E) 0.23
Intern
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Joined: 09 Oct 2018
Posts: 7
Re: A number is interesting if on adding the sum of the digits of the numb  [#permalink]

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New post 20 Aug 2019, 00:19
1. From 10 to 100 there are 91 numbers
2.
100 is not interesting, so we can set the interesting number ab
in order to be interesting:
ab=a*b+(a+b)
=> 10*a+b=a*(b+1)+b => b=9
so any number has the form a9 is interesting: 19, 29,...99
=> there are 9 numbers
So fraction is 9/91, nearly 0.1
Answer: B
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Re: A number is interesting if on adding the sum of the digits of the numb  [#permalink]

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New post 20 Aug 2019, 10:43
Hovkial wrote:
A number is interesting if on adding the sum of the digits of the number and the product of the digits of the number, the result is equal to the number. What fraction of numbers between 10 and 100 (both 10 and 100 included) is interesting?

(A) 0.01

(B) 0.1

(C) 0.11

(D) 0.16

(E) 0.23


Let ab be the number.

So a+b+a*b = 10a+b
=> 9a =a*b
=> a = 0 or b =9, a cannot be 0 as it will be single digit number then but we need numbers between 10 and 100
So, b = 9

The numbers can be 19, 29, 39, 49, 59, 69, 79, 89, 99. Total 9 numbers.
Total numbers between 10 and 100 = 100-10+1 = 91.

Fraction of numbers which are interesting = 9 / 91 ~ 0.1. So A.
GMAT Club Bot
Re: A number is interesting if on adding the sum of the digits of the numb   [#permalink] 20 Aug 2019, 10:43
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