There is a three digit number in which the third digit (units) is the square root of the first digit (hundreds). The second digit (tens) is equal to the sum of the first and third digits. The number is divisible by 2, 3, 6 and 7. What is that three digit number?

Introduction / Context:
This puzzle style question combines digit relationships with divisibility rules. We are given specific conditions relating the hundreds, tens and units digits of a three digit number and additional divisibility requirements. Such questions are designed to test structured reasoning, logical elimination and comfort with basic divisibility tests.

Given Data / Assumptions:
The number is a three digit number with digits a (hundreds), b (tens) and c (units).The third digit c is the square root of the first digit a.The second digit b is equal to a + c.The number is divisible by 2, 3, 6 and 7.Digits a, b and c are each between 0 and 9, with a not equal to zero.

Concept / Approach:
First, use the relation c^2 = a to find possible pairs for a and c. Then compute b as a + c and check that it is a single digit. This gives candidate numbers. After that, apply the divisibility rules: divisibility by 2 requires the last digit to be even, divisibility by 3 and 6 requires the sum of digits to be a multiple of 3, and divisibility by 7 must be checked by simple division.

Step-by-Step Solution:
Step 1: Because c is the square root of a, a must be a perfect square digit: 1, 4 or 9.Step 2: Possible pairs are (a, c) = (1, 1), (4, 2) or (9, 3).Step 3: For (1, 1), b = a + c = 2, so the number is 121. It is not divisible by 2, since the last digit is 1, so this fails.Step 4: For (4, 2), b = 4 + 2 = 6, giving the number 462.Step 5: For (9, 3), b = 9 + 3 = 12, which is not a single digit, so this choice is invalid.Step 6: Now test 462 for divisibility: last digit 2 shows divisibility by 2. Sum of digits is 4 + 6 + 2 = 12, which is divisible by 3, so 462 is divisible by both 3 and 6.Step 7: Finally, 462 divided by 7 equals 66 exactly, so 462 is also divisible by 7.

Verification / Alternative check:
Double checking 462, we can quickly compute 7 * 60 = 420 and 7 * 6 = 42, so 7 * 66 = 462. The sum of digits 12 confirms divisibility by 3. Since 462 is even, it satisfies the 2 and 6 divisibility rules as well. No other candidate number satisfies all conditions, confirming that 462 is unique.

Why Other Options Are Wrong:
The number 121 does satisfy the digit relations but is odd and so fails divisibility by 2 and 6. The number 943 is not constructed from the required square root relation between first and third digits and also fails the divisibility conditions. The option None of these is incorrect because 462 clearly meets all stated requirements.

Common Pitfalls:
Learners sometimes forget that b must be a single digit and may incorrectly keep the pair (9, 3). Others may only check divisibility by 2 or 3 but not by 7, leading to premature conclusions. Working systematically through each condition and eliminating impossible cases ensures a correct answer.

Final Answer:
The required three digit number is 462.