## Date X Month = Year

One way to represent April 22, 1988 is 4/22/88. Notice that the product of the month and day is equal to the two-digit representation of the year. Exactly how many times between January 1, 1990 and January 1, 2000 did the unique trait occur?
Source: mathcontest.olemiss.edu 7/23/2007

SOLUTION
1. Year 90
$90=1\times 2\times 3\times 3\times 5$
We can safely discard 1 as a factor because 1/90 means January/90 and 90 would be too large a day in January. Also, if we switch the numbers around 90/1 would make no sense because the largest month number is 12 (December). To search for the solutions of year 90, we group the four remaining factors {2, 3, 3, 5} into groups of 1-factor month number, 2-factor month number, 3-factor month number as follows.
1.1 One-factor month number
$\left (2\right )\left (3\times 3\times 5\right )$ yields 2/45 (NOT a solution) or 45/2 (NOT a solution)
$\left (3\right )\left (2\times 3\times 5\right )$ yields 3/30 (YES a solution) or 30/3 (NOT a solution)
$\left (5\right )\left (2\times 3\times 3\right )$ yields 5/18 (YES a solution) or 18/5 (NOT a solution)
1.2 Two-factor month number
$\left (2\times 3\right )\left (3\times 5\right )$ yields 6/15 (YES) or 15/6 (NO)
$\left (2\times 5\right )\left (3\times 3\right )$ yields 10/9 (YES) or 9/10 (YES)
1.3 Three-factor month number
This case yields the same result as 1.1.
1.4 Summary
For year 90 there are five solutions: 3/30, 5/18, 6/15, 9/10, and 10/9.

2. Year 91
$91=7\times 13$
2.1 One-factor month number
$\left (7\right )\left (13\right )$ yields 7/13 (YES) or 13/7 (NO)
2.2 Summary
For year 91 there is one solution: 7/13.

3. Year 92
$92=2\times 2\times 23$
3.1 One-factor month number
$\left (2\right )\left (2\times 23\right )$ yields 2/46 (NO) or 46/2 (NO)
$\left (23\right )\left (2\times 2\right )$ yields 23/4 (NO) or 4/23 (YES)
3.2 Two-factor month number
This case yields the same result as 3.1.
3.3 Summary
For year 92 there is one solution: 4/23.

4. Year 93
$93=3\times 31$
4.1 One-factor month number
$\left (3\right )\left (31\right )$ yields 3/31 (YES) or 31/3 (NO)
4.2 Summary
For year 93 there is one solution: 3/31.

5. Year 94
$94=2\times 47$
5.1 One-factor month number
$\left (2\right )\left (47\right )$ yields 2/47 (NO) or 47/2 (NO)
5.2 Summary
For year 94 there is zero solution.

6. Year 95
$95=5\times 19$
6.1 One-factor month number
$\left (5\right )\left (19\right )$ yields 5/19 (YES) or 19/5 (NO)
6.2 Summary
For year 95 there is one solution: 5/19.

7. Year 96
$96=2\times 2\times 2\times 2\times 2\times 3$
7.1 One-factor month number
$\left (2\right )\left (2\times 2\times 2\times 2\times 3\right )$ yields 2/48 (NO) or 48/2 (NO)
$\left (3\right )\left (2\times 2\times 2\times 2\times 2\right )$ yields 3/32 (NO) or 32/3 (NO)
7.2 Two-factor month number
$\left (2\times 2\right )\left (2\times 2\times 2\times 3\right )$ yields 4/24 (YES) or 24/4 (NO)
$\left (2\times 3\right )\left (2\times 2\times 2\times 2\right )$ yields 6/16 (YES) or 16/6 (NO)
7.3 Three-factor month number
$\left (2\times 2\times 2\right )\left (2\times 2\times 3\right )$ yields 8/12 (YES) or 12/8 (YES)
7.4 Four-factor month number
This case yields the same result as 7.2.
7.5 Five-factor month number
This case yields the same result as 7.1.
7.6 Summary
For year 96 there are 4 solutions: 4/24, 6/16, 8/12, and 12/8.

8. Year 97
$97=1\times 97$
8.1 Summary
For year 97 there is zero solution.

9. Year 98
$98=2\times 7\times 7$
9.1 One-factor month number
$\left (2\right )\left (7\times 7\right )$ yields 2/49 (NO) or 49/2 (NO)
$\left (7\right )\left (2\times 7\right )$ yields 7/14 (YES) or 14/7 (NO)
9.2 Two-factor month number
This case yields the same result as 9.1.
9.3 Summary
For year 98 there is one solution: 7/14.

10. Year 99
$99=3\times 3\times 11$
10.1 One-factor month number
$\left (3\right )\left (3\times 11\right )$ yields 3/33 (NO) or 33/3 (NO)
$\left (11\right )\left (3\times 3\right )$ yields 11/9 (YES) or 9/11 (YES)
10.2 Two-factor month number
This case yields the same result as 10.1.
10.3 Summary
For year 99 there are two solutions: 9/11 and 11/9.

11. Year 00 (2000)
This case yields no solution because no month multiplied by day equals 0.

12. Total
90: 5
91: 1
92: 1
93: 1
94: 0
95: 1
96: 4
97: 0
98: 1
99: 2
00: 0

Total equals 16.

Alternative solution
$90=1\times 2\times 3\times 3\times 5$
Possible months that can be formed from the prime factors are
$1,2,3,5,2\times 3=6,3\times 3=9,2\times 5=10$
The months yield the following dates
$1/90,2/45,3/30,5/18,6/15,9/10,10/9$
We discard $1/90,2/45$ because the day numbers are too big.

$91=1\times 7\times 13$
Months: $1,7$
Dates: $7/13$

$92=1\times 2\times 2\times 23$
Months: $1,2,4$
Dates: $4/23$

$93=1\times 3\times 31$
Months: $1,3$
Dates: $3/31$

$94=1\times 2\times 47$
Months: $1,2$
Dates: no solution

$95=1\times 5\times 19$
Months: $1,5$
Dates: $5/19$

$96=1\times 2\times 2\times 2\times 2\times 2\times 3$
Months: $1,2,3,4,6,8,12$
Dates: $4/24,6/16,8/12,12/8$

$97=1\times 97$
Months: $1$
Dates: no solution

$98=1\times 2\times 7\times 7$
Months: $1,2,7$
Dates: $7/14$

$99=1\times 3\times 3\times 11$
Months: $1,3,9,11$
Dates: $9/11,11/9$

The unique trait occurs $16$ times.