In the list of numbers 56, 64, 48, 90, 16 and 144, which one is the odd one out based on divisibility by 8?

Introduction / Context:
This odd one out problem is built around a simple divisibility property. The numbers 56, 64, 48, 90, 16 and 144 are all reasonably small, so it is natural to test their divisibility by a fixed integer. Spotting which numbers are or are not divisible by a chosen value quickly reveals the misfit.

Given Data / Assumptions:
The numbers provided are:

• 56
• 64
• 48
• 90
• 16
• 144

We assume that five of these numbers are divisible by 8, while one is not, and this divisibility property is the intended basis for identifying the odd one out.

Concept / Approach:
To check divisibility by 8, we can divide each number by 8 and see whether the result is an integer. If the quotient is whole with no remainder, then the number is divisible by 8. The unique number that fails this test will be the odd one out.

Step-by-Step Solution:
Step 1: Check 56. Compute 56 / 8 = 7, which is an integer, so 56 is divisible by 8.Step 2: Check 64. Compute 64 / 8 = 8, which is an integer, so 64 is divisible by 8.Step 3: Check 48. Compute 48 / 8 = 6, which is an integer, so 48 is divisible by 8.Step 4: Check 16. Compute 16 / 8 = 2, which is an integer, so 16 is divisible by 8.Step 5: Check 144. Compute 144 / 8 = 18, which is an integer, so 144 is divisible by 8.Step 6: Check 90. Compute 90 / 8 = 11.25, which is not an integer, so 90 is not divisible by 8.

Verification / Alternative check:
We can also reason through factorisations. For example, 56 = 7 * 8, 64 = 8 * 8, 48 = 6 * 8, 16 = 2 * 8 and 144 = 18 * 8. Each of these has 8 as a factor. However, 90 factors as 2 * 3 * 3 * 5 and has no factor 8, which again shows that 90 is different from the others with respect to divisibility by 8.

Why Other Options Are Wrong:
The numbers 56, 64, 48 and 16 are all clean multiples of 8, as is 144. Removing any one of them as the odd one out would still leave a group dominated by multiples of 8 while 90 remains a non multiple. Thus they cannot be considered the correct odd term.

Common Pitfalls:
Students sometimes search for complex patterns involving squares, cubes or prime factors, when a simple divisibility test would suffice. Whenever several numbers in a set are clearly multiples of a small integer, it is worth checking whether this property characterises the entire group except for one number.

Final Answer:
The only number in the list that is not divisible by 8 is 90.