You are looking for numbers that have 7 as a factor, but do not have 2,3, or 5, as a factor.
Let $A_k = \{n \in \mathbb{N}|1 \le n \le 7770\text{ and }k\text{ divides }n\}$
You are looking for:
$\begin{align*}\left|A_7 \setminus (A_2 \cap A_3 \cap A_5)\right| = & |A_7| - |A_{7\cdot 2}| - |A_{7\cdot 3}| - |A_{7\cdot 5}| \\ & + |A_{7\cdot 2\cdot 3}| + |A_{7\cdot 2\cdot 5}| + |A_{7\cdot 3\cdot 5}| \\ & - |A_{7\cdot 2\cdot 3\cdot 5}|\end{align*}$
This is because if $p,q$ are distinct primes, then $A_p\cap A_q = A_{pq}$, and also from the Inclusion/Exclusion principle. Additionally, for any prime $p$, you have:
$|A_p| = \left\lfloor \dfrac{7770}{p} \right\rfloor$
I hope you get the idea. So, the answer is:
$\dfrac{7770}{7}-\left(\dfrac{7770}{7\cdot 2} + \dfrac{7770}{7\cdot 3} + \dfrac{7770}{7\cdot 5}\right) + \left(\dfrac{7770}{7\cdot 2\cdot 3} + \dfrac{7770}{7\cdot 2\cdot 5} + \dfrac{7770}{7\cdot 3\cdot 5} \right) - \dfrac{7770}{7\cdot 2\cdot 3\cdot 5} = 296$
Edit: To verify, you can use Excel.
In cell A1, type the formula: =MOD(ROW(),2)>0
(this will be TRUE if the row number is NOT divisible by 2)
In cell B1, type: =MOD(ROW(),3)>0
C1: =MOD(ROW(),4)>0
D1: =MOD(ROW(),5)>0
E1: =MOD(ROW(),6)>0
F1: =MOD(ROW(),7)=0
(notice this one you want it to be equal to zero because that means the row number is divisible by 7)
G1: =MOD(ROW(),8)>0
H1: =MOD(ROW(),9)>0
I1: =MOD(ROW(),10)>0
J1: =AND(A1:I1)
Then select cells A1:J1 and copy the formulas down to row 7770. Then, in any open cell, add the formula: =COUNTIF(J1:J7770,TRUE) and it will tell you how many rows satisfied the conditions you set for divisibility.
Edit: To verify, you can use Excel.
In cell A1, type the formula: =MOD(ROW(),2)>0
(this will be TRUE if the row number is NOT divisible by 2)
In cell B1, type: =MOD(ROW(),3)>0
C1: =MOD(ROW(),4)>0
D1: =MOD(ROW(),5)>0
E1: =MOD(ROW(),6)>0
F1: =MOD(ROW(),7)=0
(notice this one you want it to be equal to zero because that means the row number is divisible by 7)
G1: =MOD(ROW(),8)>0
H1: =MOD(ROW(),9)>0
I1: =MOD(ROW(),10)>0
J1: =AND(A1:I1)
Then select cells A1:J1 and copy the formulas down to row 7770. Then, in any open cell, add the formula: =COUNTIF(J1:J7770,TRUE) and it will tell you how many rows satisfied the conditions you set for divisibility.[/QUOTE]
This site shows me something interesting that I did not know, can you please repeat the last part?
I'll explain myself, I have Microsoft Excel, I was able to place everything in a simple phase a - "Then select cells A1:J1 and copy the formulas down to row 7770. Then, in any open cell, add the formula: =COUNTIF(J1:J7770,TRUE) and it will tell you how many rows satisfied the conditions you set for divisibility." not so figured out
Select cells A1:J1. Right click and choose Copy (or CTRL+C on the keyboard). On the keyboard, press CTRL+G (this will bring up a dialog that says GoTo in the title. In the Reference line, put A7770. Now, while holding the Shift key (do not let go), do the following:
1. Press and release the End key
2. Press and release the Up arrow key
3. Press and release the right arrow key nine times (so you will have every cell in the range A1:J7770 highlighted).
Release the Shift key, right click, and press Paste (or CTRL+V on the keyboard).