Calculating the Day of the Week

The question asks us to determine the day of the week for April 8th, 2019, given that February 5th, 2018, was a Monday. To solve this, we need to calculate the number of days between these two dates and find the equivalent change in the day of the week.

We can approach this by first finding the day of the week exactly one year after the starting date, and then calculating the day from that point to the target date.

Step 1: Determine the Day for February 5th, 2019

From February 5th, 2018, to February 5th, 2019, is exactly one year. We need to check if this period includes a leap day (February 29th).

• The year 2018 is not a leap year ($2018 \div 4$ is not a whole number).
• The year 2019 is not a leap year ($2019 \div 4$ is not a whole number).

Since neither year is a leap year and we are crossing from 2018 into 2019 on Feb 5th, the period contains 365 days.

To find the change in the day of the week, we divide the number of days by 7 (the number of days in a week) and look at the remainder:

$\text{Number of days} = 365$

$\text{Change in day} = 365 \pmod 7$

Calculating the remainder:

$365 = 52 \times 7 + 1$

The remainder is 1. This means February 5th, 2019, will be 1 day after the day of February 5th, 2018.

Since February 5th, 2018, was a Monday, February 5th, 2019, will be Tuesday.

Step 2: Calculate the Number of Days from February 5th, 2019, to April 8th, 2019

Now we need to find the number of days from February 5th, 2019 (which is a Tuesday), to April 8th, 2019.

We sum the number of days remaining in February, plus the full month of March, plus the days in April until the 8th:

• Days remaining in February 2019: Since 2019 is not a leap year, February has 28 days. Days remaining = $28 - 5 = 23$ days.
• Days in March 2019: 31 days.
• Days in April 2019: 8 days (up to the 8th).

Total number of days $= 23 (\text{Feb}) + 31 (\text{Mar}) + 8 (\text{Apr})$

Total number of days $= 62$ days.

Step 3: Calculate the Day Offset from February 5th, 2019

We have 62 days from February 5th, 2019, to April 8th, 2019. We find the change in the day of the week by calculating the remainder when 62 is divided by 7.

$\text{Number of days} = 62$

$\text{Change in day} = 62 \pmod 7$

Calculating the remainder:

$62 = 8 \times 7 + 6$

The remainder is 6. This means April 8th, 2019, will be 6 days after the day of February 5th, 2019.

Step 4: Determine the Final Day of the Week

We know that February 5th, 2019, was a Tuesday. We need to find the day 6 days after Tuesday.

• Tuesday + 1 day = Wednesday
• Tuesday + 2 days = Thursday
• Tuesday + 3 days = Friday
• Tuesday + 4 days = Saturday
• Tuesday + 5 days = Sunday
• Tuesday + 6 days = Monday

So, 6 days after Tuesday is Monday.

Conclusion

Based on our calculations, if February 5th, 2018, was a Monday, then April 8th, 2019, will be a Monday.

Revision Table: Calendar Calculations

Additional Information: Understanding Day of Week Calculation

Calculating the day of the week for a specific date, especially across years, relies on understanding calendar cycles and the concept of the odd day or day shift.

An ordinary year has 365 days. $365 = 52 \times 7 + 1$. This means an ordinary year contains 52 full weeks and 1 extra day. So, the day of the week shifts forward by 1 day for the same date in the following year, provided no leap day is crossed.

A leap year has 366 days. $366 = 52 \times 7 + 2$. This means a leap year contains 52 full weeks and 2 extra days. If the period crosses February 29th, the day of the week shifts forward by 2 days for the same date in the following year.

In our problem, going from Feb 5, 2018, to Feb 5, 2019, crosses the period after Feb 2018 and before Feb 2019. Neither 2018 nor 2019 is a leap year, so no Feb 29th is encountered in this specific annual jump. Therefore, the shift is only +1 day.

When calculating days within the same year or consecutive years, we sum the days in each month involved and then take the total modulo 7 to find the net shift. The months have different numbers of days: 31, 28/29, 31, 30, 31, 30, 31, 31, 30, 31, 30, 31. The remainders modulo 7 for these lengths are: 3, 0/1, 3, 2, 3, 2, 3, 3, 2, 3, 2, 3. This can be a quicker way to calculate the day shift over several months.

For example, the shift from Feb 5, 2019 to Apr 8, 2019:

• Remaining Feb (23 days): $23 \pmod 7 = 2$
• March (31 days): $31 \pmod 7 = 3$
• April (8 days): $8 \pmod 7 = 1$
• Total shift: $(2 + 3 + 1) \pmod 7 = 6 \pmod 7 = 6$

This confirms our calculation of a 6-day shift from Feb 5th to Apr 8th within 2019.