A natural number when increased by 12 equals 160 times its reciprocal. Find the number.

Find a natural number whose square diminished by 84 is equal to thrice of 8 more than the given number.

The numerator of a fraction is 3 less than the denominator. If 2 is added to both the numerator and the denominator, then the sum of the new fraction and the original fraction is \( \dfrac{29}{20}\) Find the original fraction.

The sum of the squares of two consecutive even numbers is 340. Find the numbers.

The sum of the squares of two consecutive multiple of 7 is 637. Find the multiples.