* Number Properties PS Question

[a_no3mani]

Hi,

Can anyone figure out a shortcut for those 2 problems

1)How many positive integers n are there such that n + 19 is a multiple of n +3?

A) 0
B) 1
C) 2
D) 3
E) 4

2) If S= 1/11+1/12+1/13+1/14+1/15+1/16+1/17+1/18+1/19+1/20 then

A) S>1/2
B) 1/3 < S < 1/2
C) 1/6 < S < 1/3
D) S < 1/8
E) S < 1/9

Thank you

[rihanna.hayat]

Requesting instructors to check if method below is correct?

,

1)How many positive integers n are there such that n + 19 is a multiple of n +3?

A) 0
B) 1
C) 2
D) 3
E) 4

For a,b and c as +ve integers, if a is a factor of b and a is a factor of c, then a will be a factor of b+c.

Now here we have to find no. of values for n such that n+3 is a factor of n+19.
Now n+19=(n+3) + 16
So here our b=n+3 and c=16 and a=n+3,

a(=n+3) will be a factor of b(=n+3) for all values of n
but
a(=n+3) will be a factor of c(=16) for expression (n+3)X m=16, let m be a +ve integer
if m=1, n+3=16; n=13
if m=2, n+3=8; n= 5
if m=3, not possible, 16 not divisible by 3
if m=4, n+3=4; n=1
if m=5, not possible, 16 not divisible by 5
if m=6, not possible, 16 not divisible by 6
if m=7, not possible, 16 not divisible by 7
if m=8, n+3=2; n=-1

Hence for all values of m above 8, n will be negative, so no need to solve beyond.
Therefore n+3 will be a factor of n+19 for 3 values n{1, 5 and 13}

Answer (D)

Instructors please confirm.

2) If S= 1/11+1/12+1/13+1/14+1/15+1/16+1/17+1/18+1/19+1/20 then

A) S>1/2
B) 1/3 < S < 1/2
C) 1/6 < S < 1/3
D) S < 1/8
E) S < 1/9

Now we know all ten values in the equation are individually less than 1/10

Since (1/10)x 10 =1
therefore (<1/10)x 10 <1

And we know first 9 values in the equation are individually greater than 1/20

Since (1/20)x 9 =.45
therefore (>1/20)x 9 > .45
and (>1/20)x 9 + 1/20 >.45+.05
i.e (>1/20)x 9 + 1/20 > .5

Hence the S<1 and S> .5
i.e. (1/2)<S<1

Answer is (A)

Instructors please check.

[RonPurewal]

hello --

welcome to the forum. however, it appears you haven't read the forum rules (first thread in every folder).

per the forum rules, you must post the original source for every problem. (that's the original source -- not another forum or other secondhand source.) otherwise, we'll have to delete the thread so that we don't run into any copyright issues.

in addition -- also per the forum rules -- you must post a separate thread for each problem.

so:

* please post a new thread for each individual problem;

* please cite the original sources of the problems.

this thread is now locked, and will be deleted in the next week or so.
thanks.