The following formula can be used to find the day of the week on which a given year begins. First calculate
$\mathrm { a } = \mathrm { y } + \llbracket ( \mathrm { y } - 1 ) / 4 \rrbracket - \llbracket ( \mathrm { y } - 1 ) / 100 \rrbracket + \llbracket ( \mathrm { y } - 1 ) / 400 \rrbracket \text {, }$
where $y$ represents the year and $\llbracket x \rrbracket$ represents the greatest integer less than or equal to $x$. After finding $a$, find the smallest nonnegative integer $b$ such that
$a \equiv b ( \bmod 7 )$ Then $b$ gives the day of January 1, with $b = 0$ representing Sunday, $b = 1$ Monday and so on. Find the day of the week on which January 1 would occur in the year $2030 .$
A) Tuesday
B) Wednesday
C) Sunday
D) Monday

Question

Quizplus · Accepted Answer

The formula given is Zeller's congruence, which can be used to find the day of the week on which a given year begins. For the year 2030, we have:$a = 2030 + \llbracket (2030 - 1) / 4 \rrbracket - \llbracket (2030 - 1) / 100 \rrbracket + \llbracket (2030 - 1) / 400 \rrbracket$$= 2030 + \llbracket 507 \rrbracket - \llbracket 20 \rrbracket + \llbracket 5 \rrbracket$$= 2030 + 507 - 20 + 5$$= 2522$Now, we need to find the smallest nonnegative integer $b$ such that $a \equiv b ( \bmod 7 )$. We have:$2522 \equiv 1 ( \bmod 7 )$Therefore, $b = 1$, which represents Monday.

The Following Formula Can Be Used to Find the Day