Please help

[tvtt2010]

1. The function f is defined for all positive integer n by the following rule: f(n) is the number of positive integers each of which is less than n and has no common factor with n other than 1. If p is any prime number than f(p)=
A. p-1
B. p-2
C. (p+1)/2
D. (p-1)/2
E. 2

2. A certain car averages 25 mile per gallon gasoline when driven in the city and 40 miles per gallon gasoline when driven on the highway. According to these rates, which of the following is closet to number of miles per gallon that the car averages when it is driven 10 miles in the city and 50 miles on the highway?
A. 28
B. 30
C. 33
D. 36
E. 38


Thanks

[knight247]

1. F(n)= # of positive numbers less than n which have no common factor with n besides 1
Lets try a few cases
f(6)=5,4,2=3 (3 is not a part of this as 3 is a factor of 6)
f(8)=7,5,3=3 (2 and 4 is not a part of this as they are factors of 8)
f(14)=13,11,9,7,5,3=6

Since, p is prime number let us try f(n) with a few prime numbers
f(17)= 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2=15
f(11)= 10 9 8 7 6 5 4 3 2 = 9
f(13)= 12 11 10 9 8 7 6 5 4 3 2=11

From these 3 examples we can infer that for prime numbers f(n)=n-2
Hence, f(p)=p-2 Hence [spoiler]B
[/spoiler]

[tvtt2010]

Thanks for your answer, but the right answer is A. p-1
So I don't know how to do it right

[knight247]

2. In the city, the car averages 25 mpG and it is to be driven 10 miles in the city
By cross multiplying we can figure that:
If one gallon gives you 25 miles, then for 10 miles one would need 10/25=0.4 Gallons.....(1)


On the highway the car averages 40 mpG and it is to be driven 50 miles on the highway
By cross multiplying we can figure that:
If one gallon gives u 40 miles, then for 50 miles you'd need 50/40=1.25 Gallons.....(2)

Total volume of gasoline spent=0.4+1.25=1.65 Gallons
Total Distance traveled is 10 miles in the city+50 miles on the highway=60 Miles

Again, by cross multiplying if 1.65 Gallons gives you 60 miles then 1 Gallon would give you

60/1.65 miles=36.36 ≈36 Hence, D

[Anurag@Gurome]

1. The function f is defined for all positive integer n by the following rule: f(n) is the number of positive integers each of which is less than n and has no common factor with n other than 1. If p is any prime number than f(p)=

Tricky solution:
Let us take p = 2 (smallest prime)
Now number of positive integers less than p and has no common factor with p other than 1 is 1. So f(2) = 1

Only option A satisfies this result.

@knight247 : You missed 1 in all the cases.

Mathematical Approach:
Note that a prime number will have common factors other than 1 only with its multiples like p², p³ etc. As p is always greater than 1, all multiples of p are greater than p. Hence, none of the integers less than p will have any common factor with p.

Thus, f(p) = Number of positive integers less than p = (p - 1)

The correct answer is A.

[Anurag@Gurome]

2. A certain car averages 25 mile per gallon gasoline when driven in the city and 40 miles per gallon gasoline when driven on the highway. According to these rates, which of the following is closet to number of miles per gallon that the car averages when it is driven 10 miles in the city and 50 miles on the highway?

Refer to the post here : https://www.beatthegmat.com/on-the-highw ... tml#350208