This problem requires calculating the specific day of the week for a given historical date, June 17, 1998. We can solve this by calculating the total number of days passed since a known reference date and finding the remainder when divided by 7.

Calculating the Day of the Week for 17 June 1998

Step 1: Determine the Reference Day of the Week

We need a starting point. A common reference is that January 1, 1900 was a Monday. We will calculate the day of the week for January 1, 1998.

To do this, we count the number of days between January 1, 1900, and December 31, 1997. This period covers 97 full years.

• Number of years = 97.
• Number of leap years during this period (1900 is not a leap year; leap years occur every 4 years, e.g., 1904, 1908,... 1996): The count is 24 leap years.
• Total number of days = (Number of normal years * 365) + (Number of leap years * 366).
• Alternatively, Total days = (Total years * 365) + Number of leap days.
• Total days = $(97 \times 365) + 24$.

Now, we find the total day shift modulo 7. Since 365 mod 7 = 1 and 366 mod 7 = 2, we can calculate:

Total day shift = $\left( (97 \times 365) + 24 \right) \mod 7$

Using the property $a \times b \mod n = ((a \mod n) \times (b \mod n)) \mod n$ and $a + b \mod n = ((a \mod n) + (b \mod n)) \mod n$:

Total day shift = $\left( (97 \mod 7) \times (365 \mod 7) + (24 \mod 7) \right) \mod 7$

Total day shift = $\left( (97 \mod 7) \times 1 + (24 \mod 7) \right) \mod 7$

Calculating the modulo values:

• $97 = 13 \times 7 + 6$, so $97 \mod 7 = 6$.
• $24 = 3 \times 7 + 3$, so $24 \mod 7 = 3$.

Substituting these values back:

Total day shift = $\left( 6 \times 1 + 3 \right) \mod 7$

Total day shift = $\left( 6 + 3 \right) \mod 7$

Total day shift = $9 \mod 7 = 2$.

Since January 1, 1900, was a Monday, January 1, 1998, was Monday + 2 days, which is a Wednesday.

Step 2: Calculate the Number of Days Passed Until the Target Date

Now we need to calculate the number of days from January 1, 1998, to June 17, 1998. We need to sum the days in each month preceding June, plus the 17 days in June.

Total number of days passed from Jan 1, 1998, to June 17, 1998 = $31 + 28 + 31 + 30 + 31 + 17 = 168$ days.

Step 3: Calculate the Final Day of the Week

We need to find the shift in the day of the week caused by these 168 days.

Day shift = Total number of days passed $\mod 7$

Day shift = $168 \mod 7$

Since $168 = 24 \times 7$, the remainder is 0.

Day shift = $0$.

The day of the week for June 17, 1998, is the day of the week for January 1, 1998, plus the day shift.

Day for June 17, 1998 = Wednesday + 0 days = Wednesday.

Therefore, the day of the week on 17th June, 1998, was Wednesday.