Problem Solving

BTGmoderatorDC wrote: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

OA A

Source: Veritas Prep

Let us first understand the question.

The function f(77) = # of positive integers between 1 and 76, inclusive minus the integers that have common factors with the factors of 77.

Let's break 77 into prime factorization.

77 = 7*11

The multiples (less than 77) of 7 and the multiples (less than 77) of 11 would have common factors with the factors of 77; thus, we must count them and exclude.

Multiples (less than 77) of 7: 7, 14, 21, 28, 35, 42, 49, 56, 63, 70: a total of 10 multiples;
Multiples (less than 77) of 11: 11, 22, 33, 44, 55, 66: a total of 6 multiples

Thus, f(77) = 76 - 10 - 6 = 60.

The correct answer: A

Hope this helps!

-Jay
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BTGmoderatorDC wrote: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

OA A

Source: Veritas Prep

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

BTGmoderatorDC wrote: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

OA A

Source: Veritas Prep

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