Dividing 218 or 172 by the natural number n, you get a remainder of 11. Dividing n by 11, you get a remainder equal to:

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1. Since dividing 218 by n gives a remainder of 11, 218 - 11 = 207 is divisible by n :

207\equiv0\pmod{n}

2. Similarly, dividing 172 by n gives a remainder of 11, so 172 - 11 = 161 is divisible by n :

161\equiv0\pmod{n}

3. n must be a common divisor of 207 and 161. Find the greatest common divisor of 207 and 161:

- First, find the difference:

207 - 161 = 46

- Find the prime factorization of 46:

46 = 2 \times 23

- Prime factorization of 161:

161 = 7 \times 23

- Common factor is 23.

4. Therefore, the possible value of n should be 23 (since other divisions have factors that don't divide both). Now, divide n = 23 by 11:

23 \div 11 = 2 \, \text{R} \, 1

5. Thus, the remainder of dividing n by 11 is 1