Bunuel wrote:
GMAT CLUB TESTS' FRESH QUESTION:
{a, b, 1, 2}
If a and b are positive integers less than 10, what is the mode of the list above?
(1) The number of different permutations of the numbers in the list is 12.
(2) A four-digit number 21ab is divisible by 9
M36-111
Official Solution:\(\{a, b, 1, 2\}\)
If \(a\) and \(b\) are positive integers less than 10, what is the mode of the list above? The mode is the number that occurs the most frequently in a data list. For example, the mode of {2, 3, 4, 4} is 4.
(1) The number of different permutations of the numbers in the list is 12.
If the list contained 4 different numbers, the number of permutations would be \(4!=24\). Since the actual number of permutations is half of that \(\frac{4!}{2}=12\), then the list must contain two identical numbers. The list could be \(\{x, 1, 1, 2\}\), \(\{x, 2, 1, 2\}\) or \(\{x, x, 1, 2\}\) (where \(x\) is different from 1 and 2 ). So, the mode could be 1, 2, or \(a=b=x\). Not sufficient.
(2) A four-digit number \(21ab\) is divisible by 9.
An integer is divisible by 9 if the sum of its digits is divisible by 9. So, \(2+1+a+b=3+a+b\) is divisible by 9. This implies that \(a+b\) must total 6 or 15 (it cannot total more than 15, say 24, because we are told that \(a\) and \(b\) are positive integers less than 10: 1, 2, 3, 4, ..., 9). So, \(a\) and \(b\) can be: (1, 5), (5, 1), (2, 4), (4, 2), (3, 3), (6, 9), (9, 6), (7, 8), (8, 7).
If \(a\) and \(b\) are (1, 5) or (5, 1), then the mode will be 1;
If \(a\) and \(b\) are (2, 4) or (4, 2), then the mode will be 2;
If \(a\) and \(b\) are (3, 3), then the mode will be 3;
If \(a\) and \(b\) are other possible values, then there will be no mode (
if every number in the list occurs an equal number of times, then the list has no mode. For example list {1, 2, 3} has no mode).
Not sufficient.
(1)+(2) We still cannot determine which number repeats twice in the list. The list could be \(\{1, 5, 1, 2\}\), \(\{2, 4, 1, 2\}\), or \(\{3, 3, 1, 2\}\), giving the mode of 1, 2, or 3 respectively. Not sufficient.
Answer: E