If N is a positive three-digit number that is greater than 200, and each digit of N is a factor of N itself, what is the value of N?
(1) The tens digit of N is 5.
(2) The units digit of N is 5.
OAA
Please explain
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what is the value of N
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GMATGuruNY
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Statement 1:Needgmat wrote:If N is a positive three-digit number that is greater than 200, and each digit of N is a factor of N itself, what is the value of N?
(1) The tens digit of N is 5.
(2) The units digit of N is 5.
OAA
Please explain
Since each digit of N must be a factor of N, 5 must be a factor of N.
Put another way, N must be a MULTIPLE OF 5.
A multiple of 5 has a units digit of 0 or 5.
Since 0 is not a factor of any number and thus cannot be a factor of N, the units digit of N must be 5.
Since N = X55, N is ODD.
Since an odd integer has only ODD factors, and every digit of N must be a factor of N, the hundreds digit of N must be ODD.
Options for N:
355, 555, 755, 955.
Rules:
An integer is a multiple of 3 only if the sum of its digits is a multiple of 3.
An integer is a multiple of 9 only if the sum of its digits is a multiple of 9.
Since the sum of the digits of 355 is not a multiple of 3, 355 is not a multiple of 3.
Thus, the hundreds digit of 355 -- 3 -- is not a factor of 355.
Since the sum of the digits of 955 is not a multiple of 9, 955 is not a multiple of 9.
Thus, the hundreds digit of 955 -- 9 -- is not a factor of 955.
755 is not divisible by its hundreds digit of 7.
Only one option for N remains:
N=555.
SUFFICIENT.
Statement 2:
It's possible that N=555, as in Statement 1.
Since the sum of the digits of 315 is a multiple of 3, 315 is a multiple of 3.
Thus, all 3 digits of 315 -- 3, 1, and 5 -- divide into 315, implying that N=315 is a viable option.
Since N can be different values, INSUFFICIENT.
The correct answer is A.
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Hi Needgmat,
This question is thick with Number Property rules and Arithmetic patterns (knowing the Rule of 3 and the Rule of 9 would be helpful when working through this prompt).
We're told that N is a positive 3-digit number greater than 200 AND that each DIGIT of N is a FACTOR of N. We're asked for the value of N.
Fact 1: The tens digit is 5
From this Fact, we have.....
_5_ as our 3-digit number.
Since 5 MUST be a factor of this number, we would be restricted to numbers that END in 5 or 0. However, 0 is NOT a factor of ANY number, so the 3-digit number CANNOT end in 0. This leaves us with...
_55
Since we now have an odd number, the first digit CANNOT be even (since even numbers cannot divide evenly into odd numbers). We're also told that the number must be greater than 200, so there are only 4 options to consider:
355, 555, 755, 955
3 does not divide into 355
7 does not divide into 755
9 does not divide into 955
The only number that fits this description is 555.
Fact 1 is SUFFICIENT
Fact 2: The units digit of N is 5
From this Fact, we have.....
_ _5 as our 3-digit number.
From our prior work in Fact 1, we know that 555 is a possible value and that the other two digits MUST be odd.
315 is evenly divisible by 3 and 1 and 5, so this is another possible value for N.
Fact 2 is INSUFFICIENT
Final Answer: A
GMAT assassins aren't born, they're made,
Rich
This question is thick with Number Property rules and Arithmetic patterns (knowing the Rule of 3 and the Rule of 9 would be helpful when working through this prompt).
We're told that N is a positive 3-digit number greater than 200 AND that each DIGIT of N is a FACTOR of N. We're asked for the value of N.
Fact 1: The tens digit is 5
From this Fact, we have.....
_5_ as our 3-digit number.
Since 5 MUST be a factor of this number, we would be restricted to numbers that END in 5 or 0. However, 0 is NOT a factor of ANY number, so the 3-digit number CANNOT end in 0. This leaves us with...
_55
Since we now have an odd number, the first digit CANNOT be even (since even numbers cannot divide evenly into odd numbers). We're also told that the number must be greater than 200, so there are only 4 options to consider:
355, 555, 755, 955
3 does not divide into 355
7 does not divide into 755
9 does not divide into 955
The only number that fits this description is 555.
Fact 1 is SUFFICIENT
Fact 2: The units digit of N is 5
From this Fact, we have.....
_ _5 as our 3-digit number.
From our prior work in Fact 1, we know that 555 is a possible value and that the other two digits MUST be odd.
315 is evenly divisible by 3 and 1 and 5, so this is another possible value for N.
Fact 2 is INSUFFICIENT
Final Answer: A
GMAT assassins aren't born, they're made,
Rich
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A shortcut here: once you realize that S1 implies S2, you know the answer must be A, D, or E! With that in mind, start with S2 (since it's easier): if n divides by 5, it could be 515 or 555 (or whatever else), so S2 is insufficient, and we're down to A or E.
With S1, we know the number is x55. x can't be even, since no even will be a factor of an odd number, so our only possibilities are 355, 555, 755, and 955. 355 and 955 don't pass our 3 and 9 divisibility tests (3+5+5 isn't divisible by 3, 9+5+5 isn't divisible by 9), so we only need to try 555 and 755.
555 obviously works, and 755 = 700 + 49 + 6, so it ISN'T a multiple of 7. That leaves us with only 555, and the answer is A.
With S1, we know the number is x55. x can't be even, since no even will be a factor of an odd number, so our only possibilities are 355, 555, 755, and 955. 355 and 955 don't pass our 3 and 9 divisibility tests (3+5+5 isn't divisible by 3, 9+5+5 isn't divisible by 9), so we only need to try 555 and 755.
555 obviously works, and 755 = 700 + 49 + 6, so it ISN'T a multiple of 7. That leaves us with only 555, and the answer is A.
