2000 × 2000 - 2009 × 1991

= 2000 × 2000 - 2000 × 1991 - 9 × 1991

= 2000 × (2000 - 1991) - 9 × 1991

(Think of it as you have 2000 copies of 2000. For each copy, take 1991 away. You're left with 2000 copies of 9.

Or, take away 1991 copies of 2000 from 2000 copies. You're left with 9 copies of 2000)

= 2000 × 9 - 1991 × 9

(1991 × 9 = 9 × 1991, we're just rearranging)

= (2000 - 1991) × 9

(Same logic as before, you have 9 copies of 2000. For each copy, take 1991 away. You're left with 9 copies of 9

Or, take away 1991 copies of 9 from 2000 copies. You're left with 9 copies of 9 also)

= 9 × 9
= 81 (final answer)

Alternatively,

2000 × 2000 - 2009 × 1991

= 2000 × 9 + 2000 × 1991 - 2009 × 1991
= 9 × 2000 - 9 × 1991
= 9 × 9
= 81

Second alternative :

2000 × 2000 - 2009 × 1991

= 2009 × 2000 - 9 × 2000 - 2009 × 1991
= 2009 × 2000 - 2009 × 1991 - 9 × 2000
= 2009 × 9 - 2000 × 9
= 9 × 9
= 81

Secondary school method :

2000 × 2000 - 2009 × 1991

= 2000² - (2000 + 9) × (2000 - 9)

= 2000² - (2000² - 9²)

= 2000² - 2000² + 9²

= 9²

= 81

Property used : (a + b)(a - b) = a² - b²
Here, your a = 2000 and b = 9

You can use this property to solve the second question. The answer will be 246913577.

Or, use the first method (or its alternatives)

I doubt this is P5, unless "NMOS" refers to some national maths olympiad event.

Oh ya never mind, I just recall that NMOS is indeed maths olympiad, but I thought the S was counted as an E.

National Mathematical Olympiad of Singapore

I remembered seeing the logo but the S looks like the summation sign “E”

∑ is Capital/uppercase for sigma so yeah.

Thank u so much J. & Eric Nicholas K .

Oh and may I know the answer for question 8 too . Thank u in advance

It's above. 246913577.

Use the same method as before.