Finding the Number with the Most Divisors

To determine which number among the given options has the most divisors, we need to find the number of divisors for each number. The number of divisors of a positive integer can be found by first determining its prime factorization. If a number \(n\) is expressed in its prime factorization form as \(n = p_1^{a_1} \times p_2^{a_2} \times \cdots \times p_k^{a_k}\), where \(p_1, p_2, \ldots, p_k\) are distinct prime numbers and \(a_1, a_2, \ldots, a_k\) are positive integers, then the total number of divisors of \(n\) is given by the product of one more than each exponent in the prime factorization: \((a_1+1)(a_2+1)\cdots(a_k+1)\).

Calculating Divisors for Each Option

Let's find the prime factorization and the number of divisors for each of the given numbers:

Number 200

• Prime factorization of 200: \(200 = 2 \times 100 = 2 \times 10^2 = 2 \times (2 \times 5)^2 = 2 \times 2^2 \times 5^2 = 2^{1+2} \times 5^2 = 2^3 \times 5^2\).
• The exponents are 3 and 2.
• Number of divisors for 200 = \((3+1)(2+1) = 4 \times 3 = 12\).

Number 172

• Prime factorization of 172: \(172 = 2 \times 86 = 2 \times 2 \times 43 = 2^2 \times 43^1\). (Note: 43 is a prime number)
• The exponents are 2 and 1.
• Number of divisors for 172 = \((2+1)(1+1) = 3 \times 2 = 6\).

Number 156

• Prime factorization of 156: \(156 = 2 \times 78 = 2 \times 2 \times 39 = 2 \times 2 \times 3 \times 13 = 2^2 \times 3^1 \times 13^1\). (Note: 3 and 13 are prime numbers)
• The exponents are 2, 1, and 1.
• Number of divisors for 156 = \((2+1)(1+1)(1+1) = 3 \times 2 \times 2 = 12\).

Number 240

• Prime factorization of 240: \(240 = 2 \times 120 = 2 \times 2 \times 60 = 2 \times 2 \times 2 \times 30 = 2 \times 2 \times 2 \times 2 \times 15 = 2^4 \times 3 \times 5 = 2^4 \times 3^1 \times 5^1\). (Note: 3 and 5 are prime numbers)
• The exponents are 4, 1, and 1.
• Number of divisors for 240 = \((4+1)(1+1)(1+1) = 5 \times 2 \times 2 = 20\).

Comparison of Number of Divisors

Let's summarize the number of divisors for each option in a table:

Comparing the number of divisors (12, 6, 12, and 20), we can see that 240 has the highest number of divisors.

Conclusion

Based on the calculations, the number 240 has the most number of divisors among the given options.

Revision Table: Number of Divisors Concept

Additional Information: Highly Composite Numbers

Numbers with a relatively large number of divisors compared to other numbers of similar size are sometimes related to the concept of highly composite numbers. A highly composite number is a positive integer that has more divisors than any smaller positive integer. Numbers like 12, 24, 36, 48, 60, 120, 180, 240 are examples of numbers known for having many divisors relative to their size. Finding the number with the most divisors often involves comparing the exponents in their prime factorizations; numbers with many small prime factors raised to various powers tend to have more divisors.