SASMO Revolution 3

Algebra Level 2

Calculate 555 x 554555 - 554 x 555554.

169 1109 9 1089

1 solution

Nihar Mahajan
Feb 6, 2015

Let $a = 554$

Then the given expression can be written as -

$\displaystyle (a + 1)(1000a + a + 1) - a[1000(a + 1) + a]$

$\displaystyle = (a + 1)(1001a + 1) - a[1000a + 1000 + a]$

$\displaystyle = 1001a^2 + 1001a + a + 1 - 1000a^2 - 1000a - a^2$

$\displaystyle = a + a + 1$

$\displaystyle = 2a + 1$

$\displaystyle = 2\times 554 + 1$

$\displaystyle = 1108 + 1$

$\displaystyle =\boxed {1109}$

It can be done in a shorter way as follows:

Take $a=555$ and $b=554$ . The expression becomes:

$\begin{aligned}a(1000b+a)-b(1000a+b)&=&1000ab+a^2-1000ab-b^2\\&=&a^2-b^2\\ &=&(a-b)(a+b)\\&=&1\times 1109\\ &=&1109\end{aligned}$

Prasun Biswas - 6 years, 3 months ago