The numbers 42, 72, 110 and 152 are given. Three of them can be expressed as the product of two consecutive natural numbers (for example n and n + 1). Which number does not belong to this group?

Introduction / Context:
This is a classification problem involving number properties. You are given four numbers and asked to identify the one that does not share a particular structural property with the others. In this case, the property is that three of the numbers can be expressed as a product of two consecutive natural numbers, while one number cannot. Such questions help evaluate your number sense and your familiarity with simple factorizations.

Given Data / Assumptions:
The numbers presented are 42, 72, 110 and 152.
We are looking for a simple relationship such as representation as n multiplied by n plus 1 for some integer n.
Exactly three of the numbers should satisfy this relationship; one number will not and will be the odd one out.
All numbers are positive and reasonably small, making mental factorisation practical in exams.

Concept / Approach:
Two consecutive natural numbers are numbers that differ by one, such as 6 and 7, or 9 and 10. Their product is n multiplied by n plus 1. To solve this question, factor each number into its natural number factors and see whether it can be represented in the form n multiplied by n plus 1. The numbers that can be expressed this way form the main group, and the one that cannot is the odd number out.

Step-by-Step Solution:
Step 1: Factor 42. We have 42 = 6 multiplied by 7. Here, 6 and 7 are consecutive natural numbers, so 42 fits the pattern. Step 2: Factor 72. One convenient factorisation is 72 = 8 multiplied by 9. Again, 8 and 9 are consecutive natural numbers, so 72 fits the pattern. Step 3: Factor 110. We can write 110 = 10 multiplied by 11. These two numbers, 10 and 11, are consecutive integers, so 110 also fits the pattern. Step 4: Factor 152. Try factoring 152 into two integers that differ by one. The factors of 152 are 1 multiplied by 152, 2 multiplied by 76, 4 multiplied by 38 and 8 multiplied by 19. None of these factor pairs consist of consecutive numbers because 1 and 152 differ greatly, 2 and 76 differ by 74, 4 and 38 differ by 34, and 8 and 19 differ by 11. Step 5: Since no factor pair of 152 is made up of two consecutive integers, 152 cannot be written in the form n multiplied by n plus 1. Step 6: Thus, 42, 72 and 110 share the property of being products of consecutive natural numbers, while 152 does not, making 152 the odd one out.

Verification / Alternative check:
You can check for n multiplied by n plus 1 directly by trial. For n equal to 6, 6 multiplied by 7 equals 42. For n equal to 8, 8 multiplied by 9 equals 72. For n equal to 10, 10 multiplied by 11 equals 110. For n equal to 11, 11 multiplied by 12 equals 132, and for n equal to 12, 12 multiplied by 13 equals 156. The number 152 does not appear in this list or for any nearby n, confirming that it is not a product of two consecutive integers.

Why Other Options Are Wrong:
42: Equal to 6 multiplied by 7, so it clearly belongs to the main group.
72: Equal to 8 multiplied by 9, another product of consecutive numbers, so it is not the odd one.
110: Equal to 10 multiplied by 11, again fitting the pattern of consecutive factors.

Common Pitfalls:
Some candidates may mistakenly look at properties like divisibility by 3 or 7, or whether the numbers are even. Those checks do not isolate a single odd one out, since all numbers are even and share several divisibility properties. Always try to factor the numbers and look for a structural pattern, such as products of consecutive numbers, perfect squares or cubes, especially when numbers are of moderate size as in this question.

Final Answer:
The only number that is not a product of two consecutive natural numbers is 152.