Which of the following statements about numbers is/are true? I. The total number of positive factors of 72 is 12. II. The sum of the first 20 odd numbers is 400. III. The largest two digit prime number is 97.

Introduction / Context:
This question checks elementary number theory and arithmetic facts. You are given three independent statements about factors, sums of odd numbers and a prime number, and asked which statements are true. Such questions are popular in reasoning tests because they combine simple formulas with basic knowledge of prime numbers and divisors.

Given Data / Assumptions:

• Statement I: The total number of positive factors of 72 is 12.
• Statement II: The sum of the first 20 odd numbers is 400.
• Statement III: The largest two digit prime number is 97.
• We work with standard definitions of factors, odd numbers, primes and basic arithmetic formulas.

Concept / Approach:
To evaluate Statement I, we use prime factorisation and the standard formula for counting divisors. For Statement II, we recall that the sum of the first n odd numbers is n^2, a classic result which can also be checked directly. For Statement III, we examine the few two digit numbers above 90 to see which are prime and identify the largest. Once each statement has been tested, we select the option that correctly summarises which ones are true.

Step-by-Step Solution:
Check Statement I: Factorise 72 into primes. We have 72 = 2^3 × 3^2. If a number has the prime factorisation p^a × q^b, the total number of positive divisors is (a + 1)(b + 1). Here a = 3 and b = 2, so the divisor count is (3 + 1)(2 + 1) = 4 × 3 = 12. Therefore, Statement I is true. Check Statement II: The first 20 odd numbers are 1, 3, 5, ..., up to 39. It is a known result that the sum of the first n odd numbers is n^2. For n = 20, the sum is 20^2 = 400. So Statement II is true. If desired, you can verify by pairing numbers or by computing the arithmetic progression sum. Check Statement III: Two digit numbers run from 10 to 99. We need the largest prime less than 100. Numbers 98 and 96 are even, 99 is divisible by 3 (since 9 + 9 = 18 is a multiple of 3). The number 97 is not divisible by 2, 3, 5 or 7, and its square root is less than 10, so no higher small prime can divide it. Hence 97 is prime and 99, 98, 96, etc. are not. Therefore, 97 is indeed the largest two digit prime, and Statement III is true.

Verification / Alternative check:
You can quickly verify Statement II by partial sums. The sum of the first two odd numbers, 1 + 3, is 4 which is 2^2. The sum of the first three odd numbers is 1 + 3 + 5 = 9, which is 3^2. This pattern continues, supporting the formula n^2. For Statement I, you can list the factors of 72 explicitly: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36 and 72, counting 12 in total. For Statement III, checking the small set {91, 93, 95, 97, 99} confirms only 97 is prime. All three verifications support the truth of each statement.

Why Other Options Are Wrong:
Option A claims that only I and II are true, ignoring the fact that III is also true. Option B claims that only II and III are true and incorrectly discards the correct divisor count in I. Option C claims only I and III are true, incorrectly rejecting the correct formula based sum in II. Since none of these options acknowledge that all three statements hold, they are incorrect. Option D correctly recognises that I, II and III are all true.

Common Pitfalls:
Some candidates misremember the sum of first n odd numbers or mistakenly think it is n(n + 1). Others may make factorisation mistakes, for example writing 72 as 2^4 × 3 instead of 2^3 × 3^2, which leads to a wrong divisor count. In prime questions, it is common to overlook 97 and incorrectly think 99 or 95 might be prime. Carefully applying known formulas and checking divisibility by small primes avoids these errors.

Final Answer:
All three statements are correct, so the right choice is “All are true.”, which is option D.