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Problem No 1.

Solve without calculator:

999 * 999 + 1999 =

999 * 999 + 1000 + (999*1) =

999 (999 + 1 ) + 1000 =

999 (1000) + 1000 =

1000 (999 + 1) =

1000 * 1000 =

1000000

Problem No 2

Solve without calculator:

Which number is larger?

139 * 988 or 138 * 989

Solution:

(138 + 1) * 988 138 * (988 + 1)

138*988 + 988 138*988 + 138

so the answer is 139 * 988

Problem No 3

Solve without calculator:

333³ + 444³ + 555³ = ?

Step-by-step solution
Write each base as a small integer times 111:

• 333³ = (3*111)³ = 3³ × 111³
• same as in previous step 444³ = 4³ × 111³
• 555³ = 5³ × 111³

Now add them:

$$333^3 + 444^3 + 555^3 = (3^3\cdot 111^3) + (4^3\cdot 111^3) + (5^3\cdot 111^3)$$

Factor out $111^3$:

$$= 111^3\bigl(3^3 + 4^3 + 5^3\bigr)$$

Compute the small cubes:

• $3^3 = 27$
• $4^3 = 64$
• $5^3 = 125$

So

$$3^3 + 4^3 + 5^3 = 27 + 64 + 125 = 216$$

But note that $216 = 6^3$. Therefore

$$333^3 + 444^3 + 555^3 = 111^3 \cdot 216 = 111^3 \cdot 6^3 = (6\cdot 111)^3$$