+0

# MODULAR ARITHMETIC

+3
137
9

1) The last day of the school year was a Wednesday. Students then have a 77-day summer break. On what day of the week will the next school year start?

2)Grogg skip counts by 11s starting at 12. He obtains the sequence 11, 23, 34, ..., 1233. What is the remainder when the sum of all the terms in Grogg's sequence is divided by 11?

Oct 20, 2019

#1
+2

I am not that fluent in Modulus, but I will give a try,

1) Each week has 7 days. Since there are 77 days, summer break will start and end on the same day.

Summer break starts on Thursday (1)

and lasts to Thursday (77).

So the day school starts should be Friday, the next day.

2) He starts at 12, assuming you made a typo in your sequence. "11, 23, 34, ..., 1233" should be 12, 23, 34, ...., 1233.

So since he started at 12, the sum of all the terms divided by 11 will be the same as if 12 was divided by 11.

If his sequence started at 11, the sum of all the terms in the sequence will be a multiple of 11.

Since it started at 12, the remainder is

12 / 11 + 1

The remainder should be 1.

Oct 20, 2019
#2
0

#2 is incorrect.

Oct 20, 2019
#3
+1

My second attempt:

Imagine IF the sequence started at 11 instead

It would be 11, 22, 33, so on to 1232.

Notice how if you start at 12, every term in the sequence, if divided by 11, will have a remainder of 1.

So find the number of terms in teh sequence by taking 1232 (1233 subtracted by 1 so it is divisible by 11) and get 112.

There are 112 terms in the sequence. Since each term has a remainder of 1, 112 is the total of remainders.

If you take 112 and divide by 11, you get a remainder of 2.

CalculatorUser  Oct 20, 2019
edited by CalculatorUser  Oct 20, 2019
#9
0

Is my second attempt right?

CalculatorUser  Oct 20, 2019
#4
0

By the way, #1 is THURSDAY. But that was a silly mistake on your cause so I'll let it go.

Oct 20, 2019
#5
0

How is #1 thursday?...

If the last SCHOOL day is wednesday, then a 77 day summer break would start on a THURSDAY and end on a THURSDAY. That means the day after the LAST DAY of Summer will be Friday, which is the first school day of the next year.

Can someone explain this to me

CalculatorUser  Oct 20, 2019
#6
+2

77 days is 11 full weeks. Each week of summer break begins on a Thursday and ends on a Wednesday, so school begins again on Thursday.

Oct 20, 2019
#8
0

Ah, I see, my brain may have forgotten its logic tools somewhere

CalculatorUser  Oct 20, 2019
#7
+1

If you begin your Summer break on Thursday June 20, 2019[as an example], then you have:

11 days in June + 31 days in July + 31 days in August + 4 days in September =77 days. You must count the last day on September 4th, which is a Wednesday. Then the school starts on the morning of Sept 5, which is a Thursday!. You must count the first day(Thursday) AND the last day(Wednesday).

Oct 20, 2019