Question

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

Answer 1

Let the first multiple be 7n and the second multiple be 7n + 7.
Now, according to the question, we have:

\[\left( 7n \right)^2 + \left( 7n + 7 \right)^2 = 637\]

\[\Rightarrow 49 n^2 + 49 n^2 + 98n + 49 = 637\]

\[\Rightarrow 98 n^2 + 98n - 588 = 0\]

\[\Rightarrow 98\left( n^2 + n - 6 \right) = 0\]

\[\Rightarrow n^2 + 3n - 2n - 6 = 0\]

\[\Rightarrow n\left( n + 3 \right) - 2\left( n + 3 \right) = 0\]

\[\Rightarrow \left( n + 3 \right)\left( n - 2 \right) = 0\]

\[\Rightarrow n = - 3 or n = 2\]

Now, for n = −3, we have:
First number =7n = 7 × (−3) = −21
Other number = 7n + 7 = 7 × (−3) + 7 = −14
And, for n = 2, we have:
First number = 7n = 7 × 2 = 14
Other number = 7n + 7 = 7 × 2 + 7 = 21
Thus, the two numbers are either −21, −14 or 14, 21.