### Theory:

While multiplying $$2$$ integers, depending upon the sign of the two or more than two numbers, the answer can change.

1. Any one number has a negative sign:
In multiplication operation, if any one of the numbers is negative, then the result will have a negative sign $$(+ve) × (-ve) = -ve$$.
Example:
1. $265×\left(-2\right)=-530$

2. $\left(-127\right)×\left(-86\right)=10922$

3. $3×\left(-86\right)=-258$
2. Both numbers have a negative sign:
In multiplication operation, when both the numbers are negative, then the result will have a positive sign $$(-ve)×(-ve) = +ve$$. This is according to the great mathematician Euler who proved, $$(-1)×(-1)=+1$$.
Example:
1. $$(-9) × (-10) = 90$$

2. $$(-12) × (-6) = 72$$

3. $$(-10) × (-10) = 100$$

4. $$(-5) × (-5) = 25$$
3. Both numbers have positive sign:
In multiplication, when both the numbers are positive, the result will be a positive sign $$(+ve) × (+ve) =+ve$$
Example:
1. $$12 × 10 = 120$$

2. $$10 × 6 = 60$$

3. $$10 × 5 = 50$$

4. $$8 × 6 = 48$$

Multiplication with zero:
Any integer $$×$$ $$0 = 0$$. Irrespective of nature and sign of integer, when an integer is multiplied by zero, the result is $$0$$.
Example:
1. $$40 × 0 = 0$$

2. −127 $$× 0 =$$ 0