Math Problem Statement

Let n be an integer such that n ≡ 0 (mod 2), n ≡ 0 (mod 3), n ≡ 0 (mod 5), ..., n ≡ 0 (mod 23) (modulo consecutive primes). What is the smallest possible value of n satisfying the conditions above?

Solution

The problem in the image asks to find the smallest integer nn such that:

n0(mod2),n0(mod3),n0(mod5),,n0(mod23)n \equiv 0 \pmod{2}, \quad n \equiv 0 \pmod{3}, \quad n \equiv 0 \pmod{5}, \dots, \quad n \equiv 0 \pmod{23}

This means that nn must be divisible by the consecutive primes starting from 2 up to 23. Essentially, we are asked to find the least common multiple (LCM) of these primes.

The primes involved are: 2, 3, 5, 7, 11, 13, 17, 19, and 23.

Thus, the smallest possible value of nn will be the least common multiple of these primes:

n=LCM(2,3,5,7,11,13,17,19,23)n = \text{LCM}(2, 3, 5, 7, 11, 13, 17, 19, 23)

Since these are distinct prime numbers, the LCM is simply the product of all the primes:

n=2×3×5×7×11×13×17×19×23n = 2 \times 3 \times 5 \times 7 \times 11 \times 13 \times 17 \times 19 \times 23

Let me compute this for you:

n=223092870n = 223092870

Therefore, the smallest possible value of nn that satisfies all the given conditions is 223092870223092870.


Would you like further details or clarifications on this solution?

Here are 5 related questions you might find helpful:

  1. What is the significance of the least common multiple in problems like this?
  2. How do you compute the LCM for a set of numbers in general?
  3. Can this type of problem be solved using other methods like the Chinese Remainder Theorem?
  4. What happens if non-prime numbers are included in the conditions?
  5. How would the solution change if one of the primes were excluded?

Tip: When solving problems involving divisibility by several primes, finding the LCM is a straightforward and efficient method to get the smallest solution.

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Math Problem Analysis

Mathematical Concepts

Number Theory
Modular Arithmetic
Least Common Multiple

Formulas

n = LCM(2, 3, 5, 7, 11, 13, 17, 19, 23)

Theorems

Prime Factorization
Least Common Multiple

Suitable Grade Level

Grades 10-12