Remainder and Word problem

[tgf]

Hi, I found two questions on a blog, though I can't understand the authour's answer explanations https://duongminhtan.blocked/2008/0 ... tions.html


Question 1:

1.If n is a positive integer and r is the remainder when (n-1)(n+1) is divided by 24, what is the value of r?
1). n is not divisible by 2
2). n is not divisible by 3
C


Question 2:

3. Store S sold a total of 90 copies of a certain book during the seven days of last week, and it sold different numbers of copies on any two of the days. If for the seven days Store S sold the greatest number of copies on Saturday and the second greatest number of the copies on Friday, did Store S sell more than 11 copies on Friday?
1). Last week store S sold 8 copies of the book on Thursday.
2). Last week store S sold 38 copies of the book on Saturday.
B

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Question 1: Please eplain the authour's approach.

What I did:
1) Since n is Odd n+1 and n-1 is Even and two 2's in 24=2^3 * 3 gets factored away. Insufficient since 2*3 is "unaccounted for"

2) Since n isn't divisable by three, either n+1 or n-1 is divisable by three (this follows from properties of consecutive integers, right?) then the 3 gets factored away. But from the same reasoning (consecutive integers), either n-1 or n+1 is divisable by 2. So one 2 gets factored away. Still insufficient.

Taken together: Two 2's gets factored away (1) and one 2 and one 3 gets factored away (2), thus (n+1)(n-1) is divisable by all factors of 24 and the remainder is 0. Answere C.

I'm not 100% confident in my approach...


Question 2: Here I'm stuck. I have no idea why the author sums x-1 + x-2.... What's that supposed to represent?

Also I don't understand this sentance in the question: "it sold different numbers of copies on any two of the days. "

Thank you

[xcusemeplz2009]

for 2nd Q

condition given is out of seven days two days the no. of copies sold is diff than other days + sat is highest and fri is 2nd highest; that means all other day equal no of copies is sold as fri and sat are the two diff days

Statment 1): for the five days total sold is 40(8copies*5days); copies left 50=x(sat)+y(fri)[x>y]

not suff as x and y can have many values

statment 2) : sat=38 ; left with 90-38=52
now 5d(d is each days copies)+y=52
and d<y
only y=12,17,22,27.. will give i.e y>11
hence suff.

[tgf]

for 2nd Q

condition given is out of seven days two days the no. of copies sold is diff than other days + sat is highest and fri is 2nd highest; that means all other day equal no of copies is sold as fri and sat are the two diff days

Statment 1): for the five days total sold is 40(8copies*5days); copies left 50=x(sat)+y(fri)[x>y]

not suff as x and y can have many values

statment 2) : sat=38 ; left with 90-38=52
now 5d(d is each days copies)+y=52
and d<y
only y=12,17,22,27.. will give i.e y>11
hence suff.

Thanks. Sometimes the biggest challenge is just understanding the question.

[life is a test]

Question 1:

1.If n is a positive integer and r is the remainder when (n-1)(n+1) is divided by 24, what is the value of r?
1). n is not divisible by 2
2). n is not divisible by 3
C

1. if n does not divide by 2 then n = 1, i.e. it is odd, hence n-1 and n+1 must be even; 12*14 /24 then r =0 but 4*6/24 then r = 12.-->insuff

2. if n is not divisible by 3 then n+1 or n-1 must be divisible by 3; 12*14/24 then r=0 but 3*5/24 then r = 15

1. and 2.: we know that n+1 and n-1 are even and we know that one of them is divisible by 3. since the smallest even value divisible by 3 is 12 and since the other num must be even (i.e. multiple of 2) then 12*2=24 so r = 0 hence suff.

[palvarez]

Question 1:

1.If n is a positive integer and r is the remainder when (n-1)(n+1) is divided by 24, what is the value of r?
1). n is not divisible by 2
2). n is not divisible by 3

Rephrase it

Q1: What is (n-1)(n+1) (mod 24)
or
What is (n^2 -1 ) (mod 24)
or
What is n^2 (mod 24)

1. n = 1 (mod 2)
or
n^2 = 1 (mod 2)

2. n = 1, 2 (mod 3)
or
n^2 = 1 (mod 3)


Notice that 24 = 3.2^3

In order to find out a mod 24, we need to know both a (mod 2) -- because a (mod 2^3) can be computed using a mod 2-- and a (mod 3)

1 and 2 are insuff on its own.

but we know n^2 (mod 2) and n^2 (mod 3), we can find a unique value for n^2 ( mod 24). Sufficient.