Counting numbers not “connected with 6” from 1 to 60 (inclusive). A number is defined as connected with 6 if it is divisible by 6, or if the sum of its digits equals 6, or if the digit 6 appears anywhere in the number. How many integers from 1 to 60 are not connected with 6 under this definition?

Introduction / Context:
This reasoning problem tests careful counting with inclusion–exclusion style thinking. A number is “connected with 6” if it meets any of three conditions: divisible by 6, has digit sum 6, or contains the digit 6. We must count how many numbers from 1 to 60 do not meet any of these conditions and then select the correct total. This is a common set-theory counting skill used in aptitude tests and quantitative exams.

Given Data / Assumptions:

• Range of integers: 1 to 60 (inclusive).
• Connected with 6 if: (i) divisible by 6, or (ii) digit sum equals 6, or (iii) digit '6' occurs.
• We seek numbers that satisfy none of the above.

Concept / Approach:
We can mark numbers that are connected and then subtract from the total 60. Overlaps occur (for example, 24 is divisible by 6 and has digit sum 6 is false, but 36 is divisible by 6 and contains digit 6), so we must avoid double counting when making a running list. A safe approach is to enumerate systematically by property or by direct scan, ensuring each number is classified once.

Step-by-Step Solution:
Step 1: Count numbers divisible by 6 in 1..60. They are 6, 12, 18, 24, 30, 36, 42, 48, 54, 60 → 10 numbers.Step 2: Count numbers with digit sum 6: 6, 15, 24, 33, 42, 51 → 6 numbers. Some overlap with Step 1 (24, 42).Step 3: Count numbers containing the digit 6: 6, 16, 26, 36, 46, 56, and 60 → 7 numbers. Some are already counted as divisible by 6 or digit sum 6 (e.g., 6, 36, 60).Step 4: To avoid inclusion–exclusion complexity errors, the robust method is to list all numbers 1..60 and mark those that satisfy any condition. After marking, the remaining unmarked count equals numbers not connected with 6.Carrying out that classification yields 43 numbers not connected with 6.

Verification / Alternative check:
Use a quick code-like or tabulation check: scan each decade (1–10, 11–20, …, 51–60), marking divisible-by-6 values, sums-to-6 values (6, 15, 24, 33, 42, 51), and any number containing '6'. Counting unmarked values across blocks confirms 43.

Why Other Options Are Wrong:

• 18, 22, 40: These totals underestimate the many values unaffected by the three conditions.
• 42: Close but off by one; misses that among 1..60 exactly 17 are connected, leaving 43 not connected.

Common Pitfalls:

• Double counting overlap when adding property counts directly.
• Forgetting that 60 contains digit 6 and is divisible by 6.
• Missing digit-sum-equals-6 cases such as 15 or 51.

Final Answer:
43