let 'x' and 'y' represent the numbers

The sum of two numbers is 25:

x + y = 25

y = 25 - x

the sum of their squares is 313:

x^2 + y^2 = 313

plug y = 25 - x into the last equation:

x^2 + (25 - x)^2 = 313

by solving we find:

x1 = 13

x2 = 12

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for x = 13 y = 25 - 13 = 12

for x = 12 y = 25 - 12 = 13

thus the two numbers are 12 and 13.