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29 Mar 2026, 01:03
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E
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
65%
(hard)
Question Stats:
42% (01:34) correct
58%
(01:57)
wrong
based on 31
sessions
History
Date
Time
Result
Not Attempted Yet
If k is a positive integer, how many possible values are there for the greatest integer that is a divisor of both k and k + 12?
A) two
B) three
C) four
D) five
E) six
A) two
B) three
C) four
D) five
E) six
29 Mar 2026, 06:18
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This one trips a lot of people up, and I get why — the phrasing is a bit dense. Let me unpack it.
The "greatest integer that is a divisor of both k and k+12" is just gcd(k, k+12). So the question is really asking: how many distinct values can gcd(k, k+12) take?
Here's the key property: gcd(k, k+12) = gcd(k, 12). This is because any common divisor of k and k+12 must also divide their difference, which is 12. And anything that divides both k and 12 divides k+12 too. So the gcd can only be a divisor of 12.
Divisors of 12: 1, 2, 3, 4, 6, 12. That's 6 values.
Now check that each is actually achievable:
1. gcd = 1: take k = 1, then gcd(1, 13) = 1
2. gcd = 2: take k = 2, then gcd(2, 14) = 2
3. gcd = 3: take k = 3, then gcd(3, 15) = 3
4. gcd = 4: take k = 4, then gcd(4, 16) = 4
5. gcd = 6: take k = 6, then gcd(6, 18) = 6
6. gcd = 12: take k = 12, then gcd(12, 24) = 12
All 6 work. Answer is E.
Most people picking C or D probably either forget to check divisibility systematically or miss the reduction step (not realizing gcd(k, k+12) = gcd(k, 12)). Once you see that, the rest is just listing factors of 12.
The "greatest integer that is a divisor of both k and k+12" is just gcd(k, k+12). So the question is really asking: how many distinct values can gcd(k, k+12) take?
Here's the key property: gcd(k, k+12) = gcd(k, 12). This is because any common divisor of k and k+12 must also divide their difference, which is 12. And anything that divides both k and 12 divides k+12 too. So the gcd can only be a divisor of 12.
Divisors of 12: 1, 2, 3, 4, 6, 12. That's 6 values.
Now check that each is actually achievable:
1. gcd = 1: take k = 1, then gcd(1, 13) = 1
2. gcd = 2: take k = 2, then gcd(2, 14) = 2
3. gcd = 3: take k = 3, then gcd(3, 15) = 3
4. gcd = 4: take k = 4, then gcd(4, 16) = 4
5. gcd = 6: take k = 6, then gcd(6, 18) = 6
6. gcd = 12: take k = 12, then gcd(12, 24) = 12
All 6 work. Answer is E.
Most people picking C or D probably either forget to check divisibility systematically or miss the reduction step (not realizing gcd(k, k+12) = gcd(k, 12)). Once you see that, the rest is just listing factors of 12.
