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705-805 (Hard)|   Algebra|      
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Bunuel
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How many positive integers less than 1000 are 6 times the sum of their 28 Mar 2019, 06:01
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B
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Difficulty:

85% (hard)

Question Stats:

54% (02:36) correct 46% (02:15) wrong based on 279 sessions

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How many positive integers less than 1000 are 6 times the sum of their digits?

(A) 0
(B) 1
(C) 2
(D) 4
(E) 12
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B
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BrentGMATPrepNow
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How many positive integers less than 1000 are 6 times the sum of their 28 Mar 2019, 06:46
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Bunuel
How many positive integers less than 1000 are 6 times the sum of their digits?

(A) 0
(B) 1
(C) 2
(D) 4
(E) 12
Show more

Let x = the hundreds digit of the integer
Let y = the tens digit of the integer
Let z = the units digit of the integer

So, 100x + 10y + z = the VALUE of the integer
And x + y + z = the SUM of the digits

We want the VALUE of the integer to be 6 TIMES the sum of its digits
So, we want: 100x + 10y + z = 6(x + y + z)
Expand: 100x + 10y + z = 6x + 6y + 6z
Subtract 6x from both sides: 94x + 10y + z = 6y + 6z
Subtract 6y from both sides: 94x + 4y + z = 6z
Subtract z from both sides: 94x + 4y = 5z

IMPORTANT: Since x, y and z are DIGITS (from 0 to 9), we can see that x must equal zero
IF, for example, if x has a non-zero value like x = 1, we get: 94 + 4y = 5z, and there are no DIGIT values of y and z that can satisfy the equation.
So, x must be zero.

We get: 4y = 5z
Since y and z are DIGITS (from 0 to 9), we can see that there is only ONE solution: y = 5 and z = 4
So, 54 is the ONLY positive integer that satisfies the condition that the number is 6 times the sum of its digits

Answer: B

Cheers,
Brent
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How many positive integers less than 1000 are 6 times the sum of their 28 Mar 2019, 12:56
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Bunuel
How many positive integers less than 1000 are 6 times the sum of their digits?

(A) 0
(B) 1
(C) 2
(D) 4
(E) 12
Show more

three digits , x,y,z
100x+10y+z= 6*(x+y+z)
so
94x+4y=5z
x=0 so
4y=5z
y= 5 and z=4
so
54 possible no
IMO B
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How many positive integers less than 1000 are 6 times the sum of their 28 Mar 2019, 13:21
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can u explain why x was equated to zero

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How many positive integers less than 1000 are 6 times the sum of their 28 Mar 2019, 13:26
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Aderonke01
can u explain why x was equated to zero

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please see gmatprepnow solution above..
quoting below:

IMPORTANT: Since x, y and z are DIGITS (from 0 to 9), we can see that x must equal zero
IF, for example, if x has a non-zero value like x = 1, we get: 94 + 4y = 5z, and there are no DIGIT values of y and z that can satisfy the equation.
So, x must be zero.
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How many positive integers less than 1000 are 6 times the sum of their 01 Apr 2019, 06:25
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Bunuel
How many positive integers less than 1000 are 6 times the sum of their digits?

(A) 0
(B) 1
(C) 2
(D) 4
(E) 12
Show more

We can see that the integers, if they exist, are either 2 digits or 3 digits since there is no 1-digit number that is 6 times itself. If the number is 2 digits, we can create the equation:

10a + b = 6(a + b)

10a + b =6a + 6b

4a = 5b

We see that a = 5 and b = 4. If other words, the number 54 is 6 times of the sum of its digits. Notice that 6(5 + 4) = 6(9) = 54.

If the number is 3 digits, we see that the hundreds digit has to be 1. If the hundreds digit is 2 or more, we see that the quotient between the number and 6 is more than 30, but the sum of its digits is no more than 3 x 9 = 27. So we can create the equation:

100 + 10c + d = 6(1 + c + d)

100 + 10c + d = 6 + 6c + 6d

94 + 4c = 5d

We see that there is no solution for the above equation since the right hand side is at most 45 but the left hand side is at least 94.

Therefore, 54 is the only integer that is less than 1000 which is 6 times the sum of its digits.

Answer: B
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How many positive integers less than 1000 are 6 times the sum of their 06 Jan 2020, 04:08
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Bunuel
How many positive integers less than 1000 are 6 times the sum of their digits?

(A) 0
(B) 1
(C) 2
(D) 4
(E) 12

We can see that the integers, if they exist, are either 2 digits or 3 digits since there is no 1-digit number that is 6 times itself. If the number is 2 digits, we can create the equation:

10a + b = 6(a + b)

10a + b =6a + 6b

4a = 5b

We see that a = 5 and b = 4. If other words, the number 54 is 6 times of the sum of its digits. Notice that 6(5 + 4) = 6(9) = 54.

If the number is 3 digits, we see that the hundreds digit has to be 1. If the hundreds digit is 2 or more, we see that the quotient between the number and 6 is more than 30, but the sum of its digits is no more than 3 x 9 = 27. So we can create the equation:

100 + 10c + d = 6(1 + c + d)

100 + 10c + d = 6 + 6c + 6d

94 + 4c = 5d

We see that there is no solution for the above equation since the right hand side is at most 45 but the left hand side is at least 94.

Therefore, 54 is the only integer that is less than 1000 which is 6 times the sum of its digits.

Answer: B
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Can you please explain this line - with an example if possible?

If the number is 3 digits, we see that the hundreds digit has to be 1. If the hundreds digit is 2 or more, we see that the quotient between the number and 6 is more than 30, but the sum of its digits is no more than 3 x 9 = 27. So we can create the equation:
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How many positive integers less than 1000 are 6 times the sum of their 10 Jan 2020, 11:50
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Mansoor50
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Bunuel
How many positive integers less than 1000 are 6 times the sum of their digits?

(A) 0
(B) 1
(C) 2
(D) 4
(E) 12

We can see that the integers, if they exist, are either 2 digits or 3 digits since there is no 1-digit number that is 6 times itself. If the number is 2 digits, we can create the equation:

10a + b = 6(a + b)

10a + b =6a + 6b

4a = 5b

We see that a = 5 and b = 4. If other words, the number 54 is 6 times of the sum of its digits. Notice that 6(5 + 4) = 6(9) = 54.

If the number is 3 digits, we see that the hundreds digit has to be 1. If the hundreds digit is 2 or more, we see that the quotient between the number and 6 is more than 30, but the sum of its digits is no more than 3 x 9 = 27. So we can create the equation:

100 + 10c + d = 6(1 + c + d)

100 + 10c + d = 6 + 6c + 6d

94 + 4c = 5d

We see that there is no solution for the above equation since the right hand side is at most 45 but the left hand side is at least 94.

Therefore, 54 is the only integer that is less than 1000 which is 6 times the sum of its digits.

Answer: B

Can you please explain this line - with an example if possible?

If the number is 3 digits, we see that the hundreds digit has to be 1. If the hundreds digit is 2 or more, we see that the quotient between the number and 6 is more than 30, but the sum of its digits is no more than 3 x 9 = 27. So we can create the equation:
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For a three-digit integer, the maximum value for the sum of the digits is 27 (which is the sum of the digits of the three-digit number 999). Thus, for any three-digit integer, the maximum value of 6 times the sum of the digits is 6 x 27 = 162. Even the smallest possible three-digit integer with a hundreds digit of 2 (which is 200) exceeds 162; that’s why, if 6 times the sum of the digits is to equal the number, the hundreds digit must be 1.
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How many positive integers less than 1000 are 6 times the sum of their 23 Jul 2026, 12:12
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lets break our approach in three parts:
1. from 1 to 9--not applicable
2. from 10 to 99
10A+B=6(A+B)
4A=5B hence only no. possible in this range is 54.
3.from 100 to 999
100A+10B+C=6A+6B+6C
94A+4B=5C where A,B,C are digits from 0 to 9.
so this equation doesn't seems viable
hence correct choice is 1.
B
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