Welcome back to our series on number properties. In the last article, we introduced remainders. This time we will address my favorite use of remainders on GMAT quant: day of the week problems. Have you ever encountered a problem like this?
June 25, 1982, fell on a Friday. On which day of the week did June 25, 1987, fall? (Note: 1984 was a leap year).
Many test-takers are not excited to see days of the week as answer choices on the quantitative portion of a test. What’s going on here?
To deal with this problem, we need to consider two facts that every human should know: there are 7 days in a week, and there are 365 days in a year. So if we divide 365 by 7, the remainder tells us how many “extra days” occur each year.
365 / 7 → 7 * 52 = 364 → 365 – 364 = 1
Each year is made up of 52 weeks and then 1 “extra day” (leap years have two extra days).
If the first day of the year, Jan 1, is a Sunday, the second-to-last day of the year, Dec 30, will be a Saturday, closing out the 52nd week of the year.
The last day of the year will be a Sunday again, and the first day of the next year, Jan 1, will be a Monday.
To simplify all this, since 365/7 has remainder 1, a given date “rotates” to the next day of the week each normal year. Since 366/7 has remainder 2, a given date “rotates” two days of the week each leap year.
So to solve these day-of-the-week problems, add up the number of normal years and the number of leap years occurring between the start date and the end date. We are going from 1982 to 1987 for a total of 5 years, one of which is a leap year. So there will be 4 normal “rotations” for the next day of the week and one “double rotation” for the leap year, for a total of 6 “rotations.” This means that in 1987, June 25 will fall on the day of the week immediately before the day on which it fell in 1982, moving from Friday to Thursday. The correct answer is E.
Here’s one more official problem for practice:
November 16, 2001, was a Friday. If each of the years 2004, 2008, and 2012 had 366 days, and the remaining years from 2001 through 2014 had 365 days, what day of the week was November 16, 2014?
This problem avoids the term “leap year” and reminds us how many days occurred in each year in the time span. We will follow the same process as before.
From 2001 to 2014 is a total of 13 years. Ten of these years are normal, and three of them are leap years. The day of the week for November 16 “rotates” one day each normal year and two days each leap year, for a total of 16 “rotations.”
We can use remainders one more time to finish answering this question. Every 7 “rotations” is one week, so we are only concerned about the remainder of 16/7. This remainder is 2, so from 2001 to 2014, November 16 moved two days of the week from Friday to Sunday. The correct answer is A.
Now you’re prepared for day-of-the-week problems on GMAT quant. Just remember, in normal years, a given date rotates to the next day of the week. And leap years, it rotates two days. The next two articles in the series will address the related topics of greatest common factors and least common multiples.
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Contributor: Elijah Mize (Apex GMAT Instructor)