How many of the integers 1, 2, … , 120, are divisible by none of 2, 5 and 7?

The correct option is 41.

To find the number of integers from 1 to 120 that are divisible by none of 2, 5, and 7, we can use the Principle of Inclusion-Exclusion (PIE).

First, let $S$ be the set of integers from 1 to 120. The total number of elements in this set is:
$\left|S\right|=120$

Let us define the following sets:
• $A$: Set of integers in $S$ divisible by 2.
• $B$: Set of integers in $S$ divisible by 5.
• $C$: Set of integers in $S$ divisible by 7.

We calculate the number of elements in each of these sets using the floor function (represented by $\lfloor \rfloor$):

$\left|A\right|=\lfloor \frac{120}{2}\rfloor =60$

$\left|B\right|=\lfloor \frac{120}{5}\rfloor =24$

$\left|C\right|=\lfloor \frac{120}{7}\rfloor =17$

Next, we determine the number of integers divisible by combinations of two of these prime numbers (which correspond to the intersections of the sets):

• Divisible by both 2 and 5 (i.e., divisible by 10):
$|A\cap B|=\lfloor \frac{120}{10}\rfloor =12$

• Divisible by both 5 and 7 (i.e., divisible by 35):
$|B\cap C|=\lfloor \frac{120}{35}\rfloor =3$

• Divisible by both 2 and 7 (i.e., divisible by 14):
$|A\cap C|=\lfloor \frac{120}{14}\rfloor =8$

Now, we find the number of integers divisible by all three numbers (divisible by 2, 5, and 7, which means divisible by 70):

$|A\cap B\cap C|=\lfloor \frac{120}{70}\rfloor =1$

According to the Principle of Inclusion-Exclusion, the number of integers divisible by at least one of 2, 5, or 7 is given by:

$|A\cup B\cup C|=\left|A\right|+\left|B\right|+\left|C\right|-\left(\right|A\cap B|+|B\cap C|+|A\cap C\left|\right)+|A\cap B\cap C|$

Substituting our calculated values into the formula:

$|A\cup B\cup C|=60+24+17-(12+3+8)+1$

$|A\cup B\cup C|=101-23+1=79$

Thus, there are 79 integers between 1 and 120 that are divisible by at least one of 2, 5, or 7.

Finally, to find the number of integers divisible by none of these numbers, we subtract this quantity from the total number of integers in the set:

$\text{Number of required integers}=\left|S\right|-|A\cup B\cup C|$

$\text{Number of required integers}=120-79=41$