Hi All,
We're told that we have 7 CONSECUTIVE positive integers. We're asked how many of them are divisible by 6. To start, with 7 consecutive integers, only 1 or 2 of those integers will be divisible by 6. Once you have a number that is divisible by 6, the 'next' number that is divisible by 6 will either be '6 more' or '6 less' than the original number. For example: 0, 6, 12, 18, 24, etc.
1) Their average is divisible by 6.
With 7 consecutive integers, the AVERAGE will equal the "middle" number (in this case, the 4th number), so we can use a bit of 'brute force' to determine whether a pattern exists here or not.
IF... the numbers are...
3, 4, 5, 6, 7, 8, 9 .... there's 1 number that is divisible by 6.
9, 10, 11, 12, 13, 14, 15 .... there's 1 number that is divisible by 6.
15, 16, 17, 18, 19, 20, 21 .... there's 1 number that is divisible by 6.
Etc.
There will ALWAYS be just 1 number that is divisible by 6.
Fact 1 is SUFFICIENT
2) Their median is divisible by 12
With 7 consecutive integers, the AVERAGE = MEDIAN and both values will equal the "middle" number (in this case, the 4th number), so we can use a bit of 'brute force' here too. The difference is that the 4th value will have to be a multiple of 12.
IF... the numbers are....
9, 10, 11, 12, 13, 14, 15 .... there's 1 number that is divisible by 6.
21, 22, 23, 24, 25, 26, 27 .... there's 1 number that is divisible by 6.
Etc.
There will ALWAYS be just 1 number that is divisible by 6.
Fact 2 is SUFFICIENT
Final Answer:
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