The function f(a) is defined for all positive integers a as the number

Question

The function f(a) is defined for all positive integers a as the number of positive integers which are less than a and have no common factor with a other than 1. What is f(77)?

A. 60
B. 63
C. 66
D. 70
E. 76

BrentGMATPrepNow · Accepted Answer

Nice question!

Since 77 = (11)(7), any positive multiple of 7 or 11 will have a common factor with 77 where that common factor is greater than 1. 
For example, 77 and 14 share a common factor of 7. 
And 77 and 35 share a common factor of 7. 
Likewise, 77 and 55 share a common factor of 11.
And 77 and 33 share a common factor of 11.

Conversely, 77 and 10 do NOT share a common factor (other than a common factor of 1).
And 77 and 19 do NOT share a common factor (other than a common factor of 1).

There are   76   positive integers that are LESS THAN 77
The multiples of 7 that are less than 77 are as follows:  7, 14, 21, 28, . . . . 63, 70 (TOTAL = 10)  
The multiples of 11 that are less than 77 are as follows:  11, 22, 33, . . . 55, 66 (TOTAL = 6)

So, f(77) =   76   - ( 10  +  6 )
= 76 - 16
= 60

Answer:  A

Cheers,
Brent

pushpitkc · Answer

f(77) contains numbers less than 77, which have no common factors with 77 other than 1.

The multiples of 7 and 11 will contain factors that are common.

The number 7 contains 10 such numbers from 7 till 70(7,14,21,28,35,42,49,56,63,70)
Similarly, the number 11 contains 6 numbers from 11 to 66(11,22,33,44,55,66)

Therefore, the value of f(77) is 76 - 10 - 6 = 60(Option A)

sd260487 · Answer

Factors of 77=1, 7, 11, 77
+ive integers, which are <77 = 76
Exclusion from 'a' = 7, 14,..., 70 (10 numbers) + 11, 22,...,66 (6 numbers)
Answer: f(a) = 76-10-6 = 60
Choice: A

ScottTargetTestPrep · Answer

We are given that the function f(a) is defined for all positive integers a as the number of positive integers that are less than a and have no common factor with a other than 1, and we need to determine f(77).

We can prime factor 77 as 7 x 11. So now we can express the question as: How many positive integers less than 77 do not have 7 or 11 as factors? Our first step is to determine the number of multiples of 7 and 11 that are less than 77.

For 11: 11, 22, 33, 44, 55, 66 = 6 multiples

For 7: 7, 14, 21, 28, 35, 42, 49, 56, 63, 70 = 10 multiples

We can see that there are 6 + 10 = 16 integers that have common factors (other than 1) with 77. Since there are 76 positive integers less than 77 and 16 of them have common factors (other than 1) with 77, there are 76 - 16 = 60 numbers that have no common factors (other than 1) with 77.

Answer: A

LoneSurvivor · Answer

We need to find the primes less than 77.
Now 77=7*11
so, No of primes will be 77*(1-1/7)*(1-1/11)=60

KarishmaB · Answer

We need to find number of positive integers less than 77 with no common factor with 77 (other than 1).
Factors of 77 are 1, 7, 11 and 77. 
So out of the numbers 1 - 76, we need to remove those numbers which have 7 or 11 as a factor. 
Since 7 * 11 = 77, in 1 - 76, 10 numbers will have a factor of 7. 
Since 11 * 7 = 77, in 1 - 76, 6 numbers will have a factor of 7.

Removing these 10 + 6 numbers out of 76 numbers, we get 60 valid numbers.
Note that no number will have both 7 and 11 as factors since the first such number is 77.

Answer (A)

pierjoejoe · Answer

f(a) = all the co-primes that are less than a and greater than zero

a co-prime is a number such that LCM(a; K) = 1

f(77) --> we have 76 values less than 77

all the multiples of either 7 or 11 will have a LCM that will be either 7 or 11. so we have to remove all of those --> how many multiple of 7 are there between 7 and 76? --> (70-7)/7 +1 = 10

how many multiples of 11? (66-11)/11 +1 = 6

total = 76-10-6 = 60

bumpbot · Answer

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The function f(a) is defined for all positive integers a as the number

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