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**

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**

yields 2/45 (NOT a solution) or 45/2 (NOT a solution)

yields 3/30 (YES a solution) or 30/3 (NOT a solution)

yields 5/18 (YES a solution) or 18/5 (NOT a solution)

**1.2 Two-factor month number**

yields 6/15 (YES) or 15/6 (NO)

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
**

**2.1 One-factor month number**

yields 7/13 (YES) or 13/7 (NO)

**2.2 Summary**

For year 91 there is one solution: 7/13.

**3. Year 92
**

**3.1 One-factor month number**

yields 2/46 (NO) or 46/2 (NO)

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
**

**4.1 One-factor month number**

yields 3/31 (YES) or 31/3 (NO)

**4.2 Summary**

For year 93 there is one solution: 3/31.

**5. Year 94
**

**5.1 One-factor month number**

yields 2/47 (NO) or 47/2 (NO)

**5.2**S

**ummary**

For year 94 there is zero solution.

**6. Year 95
**

**6.1 One-factor month number**

yields 5/19 (YES) or 19/5 (NO)

**6.2 Summary**

For year 95 there is one solution: 5/19.

**7. Year 96
**

**7.1 One-factor month number**

yields 2/48 (NO) or 48/2 (NO)

yields 3/32 (NO) or 32/3 (NO)

**7.2 Two-factor month number**

yields 4/24 (YES) or 24/4 (NO)

yields 6/16 (YES) or 16/6 (NO)

**7.3 Three-factor month number**

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
**

**8.1 Summary**

For year 97 there is zero solution.

**9. Year 98
**

**9.1 One-factor month number**

yields 2/49 (NO) or 49/2 (NO)

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
**

**10.1 One-factor month number**

yields 3/33 (NO) or 33/3 (NO)

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.

**Answer**: 16

**Alternative solution**

Possible months that can be formed from the prime factors are

The months yield the following dates

We discard because the day numbers are too big.

Months:

Dates:

Months:

Dates:

Months:

Dates:

Months:

Dates: no solution

Months:

Dates:

Months:

Dates:

Months:

Dates: no solution

Months:

Dates:

Months:

Dates:

The unique trait occurs times.

Is there an easier way to find out how many times this occurs from 1/1/00 to 12/31/99?

Please refer to the much simpler alternative solution.