Question

The sum of the squares of two consecutive multiples of 7 is 637. The product of two multiples is:

Answer 1

Multiples of 7 are 7, 14, 21, 28...

Here the difference between two consecutive multiples will be 7

In general, the difference between two consecutive multiples of $n$ will be $n$

Let two consecutive numbers be $x$ and $x+7$

Given that sum of the squares of two consecutive multiples is 637

$\Rightarrow x^2+(x+7{)}^2=637$

$\Rightarrow$ $x^2+x^2+14x+49=637$

$\Rightarrow$ $2x^2+14x+49-637=0$

$\Rightarrow$ $2x^2+14x-588=0$

$\Rightarrow$ $2(x^2+7x-294)=0$

$\Rightarrow$ $x^2+7x-294=0$

$\Rightarrow$ $x^2+21x-14x-294=0$

$\Rightarrow$ $x(x+21)-14(x+21)=0$

$\Rightarrow$ $(x+21)(x-14)=0$

$\Rightarrow$ $x+21=0$ or $x-14=0$

$\Rightarrow$ $x=-21$ or $x=14$

Case(1)

If one number is $14$

Other, number $=x+7=14+7=21$

So The product of two multiple $14\times 21=294$

Case(2)

If one number is $-21$

Other, number $=x+7=-21+7=-14$

So The product of two multiple $-14\times -21=294$

In both the cases, the answer is 294