If we write November 8,1988 as 8.11.88, we see `8 xx 11 = 88.` How many such days are in `1972` ?

Options

A. 6
B. 4
C. 3
D. 2

Answer

To solve the problem of how many days in the year 1972 can be expressed in the format \( x \cdot y = 72 \), where \( x \) is the day and \( y \) is the month, we can follow these steps:

### Step 1: Understand the Format
We need to express the date in the format \( x.y.72 \) where \( x \) is the day and \( y \) is the month. The equation we are looking for is \( x \cdot y = 72 \).

### Step 2: Factorize 72
We need to find pairs of integers \( (x, y) \) such that \( x \cdot y = 72 \). First, we will find the factors of 72.

The prime factorization of 72 is:
\[ 72 = 2^3 \cdot 3^2 \]

The factors of 72 are:
1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72.

### Step 3: Find Valid Pairs
Now we will find pairs \( (x, y) \) from the factors of 72 where \( x \) (day) can be between 1 and 31 and \( y \) (month) can be between 1 and 12.

- \( (1, 72) \) - Invalid (72 is not a month)
- \( (2, 36) \) - Invalid (36 is not a month)
- \( (3, 24) \) - Valid
- \( (4, 18) \) - Valid
- \( (6, 12) \) - Valid
- \( (8, 9) \) - Valid
- \( (9, 8) \) - Valid
- \( (12, 6) \) - Valid
- \( (18, 4) \) - Invalid (18 is not a month)
- \( (24, 3) \) - Invalid (24 is not a month)
- \( (36, 2) \) - Invalid (36 is not a month)
- \( (72, 1) \) - Invalid (72 is not a month)

### Step 4: List Valid Days
The valid pairs that satisfy \( x \cdot y = 72 \) and the constraints on \( x \) and \( y \) are:
1. \( (3, 24) \) - 24th March
2. \( (4, 18) \) - 18th April
3. \( (6, 12) \) - 12th June
4. \( (8, 9) \) - 9th August
5. \( (9, 8) \) - 8th September
6. \( (12, 6) \) - 6th December

### Step 5: Count the Valid Days
From the valid pairs, we can see that there are a total of 6 valid days in the year 1972.

### Final Answer
Thus, the total number of such days in 1972 is **6**.

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