In this problem you are asked to go through the first steps of the Sieve of Eratosthenes for the integers from 2310 to 2340.
[Hint: 2310 is divisible by 2,3, and 5.]
Check all numbers that are multiples of 2, 3, or 5.
A. 2310
B. 2311
C. 2312
D. 2313
E. 2314
F. 2315
G. 2316
H. 2317
I. 2318
J. 2319
K. 2320
L. 2321
M. 2322
N. 2323
O. 2324
P. 2325
Q. 2326
R. 2327
S. 2328
T. 2329
U. 2330
V. 2331
W. 2332
X. 2333
Y. 2334
Z. 2335
AA. 2336
AB. 2337
AC. 2338
AD. 2339
AE. 2340

To implement the Sieve of Eratosthenes for numbers from 2310 to 2340, we need to mark any number divisible by 2, 3, or 5. Here's a step-by-step solution:

1. **Identify the Numbers Divisible by 2**: Any even number is divisible by 2.

2. **Identify the Numbers Divisible by 3**: A number is divisible by 3 if the sum of its digits is divisible by 3.

3. **Identify the Numbers Divisible by 5**: A number is divisible by 5 if it ends in 0 or 5.

After checking each number between 2310 and 2340, here is the result:

- **Divisible by 2**: 2310, 2312, 2314, 2316, 2318, 2320, 2322, 2324, 2326, 2328, 2330, 2332, 2334, 2336, 2338, 2340
- **Divisible by 3**: 2310, 2313, 2316, 2319, 2322, 2325, 2328, 2331, 2334, 2337, 2340
- **Divisible by 5**: 2310, 2315, 2320, 2325, 2330, 2335, 2340

### Marked Numbers:
Combining these lists, the numbers divisible by 2, 3, or 5 are:
**2310, 2312, 2313, 2314, 2315, 2316, 2318, 2319, 2320, 2322, 2324, 2325, 2326, 2328, 2330, 2331, 2332, 2334, 2335, 2336, 2337, 2338, 2340**

These are the numbers that are multiples of 2, 3, or 5 in the range from 2310 to 2340.

Would you like further explanation of any step? Here are five related questions to extend this topic:

1. How does the Sieve of Eratosthenes work for identifying prime numbers?
2. What is the general pattern to check for divisibility by 7 or 11?
3. How could we modify this approach for a larger range, like up to 100,000?
4. What are the efficiency benefits of using the Sieve of Eratosthenes over trial division?
5. How does the Sieve of Eratosthenes relate to modern computational number theory?

**Tip**: The Sieve of Eratosthenes is particularly efficient for finding all primes up to a large \( n \) because it avoids repeated division tests by systematically marking non-prime numbers.

In this problem you are asked to go through the first steps of the Sieve of Eratosthenes for the integers from 2310 to 2340.
[Hint: 2310 is divisible by 2,3, and 5.]
Check all numbers that are multiples of 2, 3, or 5.
A. 2310 B. 2311 C. 2312 D. 2313 E. 2314 F. 2315 G. 2316 H. 2317 I. 2318 J. 2319 K. 2320 L. 2321 M. 2322 N. 2323 O. 2324 P. 2325 Q. 2326 R. 2327 S. 2328 T. 2329 U. 2330 V. 2331 W. 2332 X. 2333 Y. 2334 Z. 2335 AA. 2336 AB. 2337 AC. 2338 AD. 2339 AE. 2340

Learn how to apply the Sieve of Eratosthenes for integers 2310 to 2340 by checking divisibility by 2, 3, and 5. This exercise covers key concepts in number theory, including divisibility rules and the Sieve of Eratosthenes, suitable for students in Grades 8-10.

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