What are the two natural numbers whose squares add up to 52?

Introduction / Context:
This question tests the ability to work with sums of squares of natural numbers. It is a typical number theory style problem that can be solved either by trial of small squares or by systematic reasoning about possible square values.

Given Data / Assumptions:

• We need two natural numbers, say m and n.
• The condition is m^2 + n^2 = 52.
• Both m and n are positive integers.

Concept / Approach:
We look for pairs of natural numbers whose squares sum to 52. Since 52 is not very large, we can try small squares: 1, 4, 9, 16, 25, 36, and so on, and see which pair of these sums to 52. We must also ensure that the pairs match one of the options.

Step-by-Step Solution:
Step 1: List small squares of natural numbers: 1^2 = 1, 2^2 = 4, 3^2 = 9, 4^2 = 16, 5^2 = 25, 6^2 = 36, 7^2 = 49.Step 2: Look for two squares that sum to 52.Step 3: Check 16 + 36 = 52, which corresponds to 4^2 + 6^2 = 52.Step 4: Confirm that there is no other pair of squares in this range that sums to 52: 25 + 25 = 50, 9 + 36 = 45, 1 + 49 = 50, and so on.Step 5: Therefore, the only natural numbers whose squares add to 52 are 4 and 6.

Verification / Alternative check:
We can verify by direct substitution: 4^2 + 6^2 = 16 + 36 = 52. For completeness, we may quickly test the options: 2 and 7 give 4 + 49 = 53, 3 and 5 give 9 + 25 = 34, and 5 and 6 give 25 + 36 = 61. Only 4 and 6 satisfy the condition.

Why Other Options Are Wrong:

• 2, 7: 2^2 + 7^2 = 4 + 49 = 53, not 52.
• 3, 5: 3^2 + 5^2 = 9 + 25 = 34.
• 5, 6: 5^2 + 6^2 = 25 + 36 = 61.

Common Pitfalls:
Some students may mis square numbers or add incorrectly. It is also possible to overlook a valid pair if the list of squares is not written systematically. Writing all small squares clearly and checking each combination methodically avoids such mistakes.

Final Answer:
The two natural numbers whose squares sum to 52 are 4 and 6.