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GRE Prep Club Team Member
Joined: 20 Feb 2017
Posts: 2508
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GPA: 3.39
A positive integer is divisible by 3 if and only if the sum of its dig
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29 Nov 2021, 07:25
29 Nov 2021, 07:25
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Question Stats:
60% (01:44) correct
40% (01:32) wrong based on 5 sessions
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A positive integer is divisible by 3 if and only if the sum of its digits is divisible by 3. If the six-digit integer is divisible by 3, and n is of the form 1k2,k24, where k represents a digit that occurs twice, how many values could n have?
A 2
B. 3
C. 4
D. 5
E 10
A 2
B. 3
C. 4
D. 5
E 10
Re: A positive integer is divisible by 3 if and only if the sum of its dig
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01 Dec 2021, 02:17
01 Dec 2021, 02:17
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Divisibility by 3 can be easily checked without adding all the digits of a large number:
Say the number is 36275981720160
Now rather than adding all the digits, simply ignore digits if they are multiples of 3. Also ignore some digits if their sum is a multiple of 3.
What I mean is this: Go one digit at a time
3 - Ignore
6 - Ignore
2 + 7 = 9 - Ignore
5 - Hold
9 - Ignore
8 + 1 - Ignore
7 + 2 - Ignore
1 + 5 (which you were holding) = 6 - Ignore
6 - Ignore
0 - Ignore
You are not holding anything so the number is divisible by 3.
Similarly, when you have a number such as 1k2k24,
1+2 = 3 - Ignore
2+4 = 6 - Ignore
So if k is a multiple of 3, then the number will be divisible by 3, else it will not be. There are 4 single digit multiples of 3: 0, 3, 6, 9
Hence k can take 4 values.
Say the number is 36275981720160
Now rather than adding all the digits, simply ignore digits if they are multiples of 3. Also ignore some digits if their sum is a multiple of 3.
What I mean is this: Go one digit at a time
3 - Ignore
6 - Ignore
2 + 7 = 9 - Ignore
5 - Hold
9 - Ignore
8 + 1 - Ignore
7 + 2 - Ignore
1 + 5 (which you were holding) = 6 - Ignore
6 - Ignore
0 - Ignore
You are not holding anything so the number is divisible by 3.
Similarly, when you have a number such as 1k2k24,
1+2 = 3 - Ignore
2+4 = 6 - Ignore
So if k is a multiple of 3, then the number will be divisible by 3, else it will not be. There are 4 single digit multiples of 3: 0, 3, 6, 9
Hence k can take 4 values.
gmatclubot
Re: A positive integer is divisible by 3 if and only if the sum of its dig [#permalink]
01 Dec 2021, 02:17
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