Since y is a natural number, y ≠ 0, which gives y = 5. Substituting y = 5 into x = 2y yields x = 10. Checking the conditions: the sum of squares is 10² + 5² = 125, the difference is 10 - 5 = 5, and 25 × 5 = 125. Also, the sum is 10 + 5 = 15 = 3 × 5. Both conditions are satisfied.

Since y is a natural number, y ≠ 0, which gives y = 5. Substituting y = 5 into x = 2y yields x = 10. Checking the conditions: the sum of squares is 10² + 5² = 125, the difference is 10 - 5 = 5, and 25 × 5 = 125. Also, the sum is 10 + 5 = 15 = 3 × 5. Both conditions are satisfied.

find two natural numbers the sum of 4 squares is 25 times the sum and also equal to 3 times the difference

Answer: The two natural numbers are 10 and 5.

Step-by-step solution

Step 1: Analyze the conditions and formulate the system

Let the two natural numbers be $x$ and $y$ such that $x > y$ . The prompt contains typographical ambiguity: 'the sum of 4 squares is 25 times the sum and also equal to 3 times the difference'. In classic algebra problems, this reads as: 'the sum of their squares is $25$ times their difference, and the sum of the numbers is $3$ times their difference' or $x^2 + y^2 = 25(x + y)$ and related conditions.

Step 2: Examine the linear relationship between sum and difference

In standard problems of this type, the sum of the numbers is $3$ times their difference: $x + y = 3(x - y)$ . Expanding this gives $x + y = 3x - 3y$ , which simplifies directly to $2x = 4y$ , meaning $x = 2y$ .

Step 3: Substitute into the sum of squares equation

Using the relation that the sum of their squares is $25$ times their difference, we substitute $x = 2y$ . This gives $(2y)^2 + y^2 = 25(2y - y)$ , which simplifies to $4y^2 + y^2 = 25y$ , or $5y^2 = 25y$ .

Step 4: Solve for the two natural numbers

Since $y$ is a natural number, $y \neq 0$ , which gives $y = 5$ . Substituting $y = 5$ into $x = 2y$ yields $x = 10$ . Checking the conditions: the sum of squares is $10^2 + 5^2 = 125$ , the difference is $10 - 5 = 5$ , and $25 \times 5 = 125$ . Also, the sum is $10 + 5 = 15 = 3 \times 5$ . Both conditions are satisfied.

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