Question

pls answer and I will mark as brainliest​

Answer 1

Topic

• Base conversion

Understanding the Problem

① The 1 in each digit has a place value of the power of the base. The exponent of the numbers is in descending order.

② The conversion of the base can be done by repeated division.

③ Let the quotient be the greatest digit and list the remainders in ascending order.

Solution

(1)

$\implies (175)_{10}$

Divide by 4,

⇒ Quotient 43, remainder 3.

Dividing the quotient by 4.

⇒ Quotient 10, remainder 3.

Again,

⇒ Quotient 2, remainder 2.

$\implies \boxed{(2233)_{4}}$

(2)

$\implies (132)_{4}$

In base 10,

$\implies 4^{2}\cdot 1+3\cdot 4^{1}+2\cdot 4^{0}$

$\implies 16+12+2$

$\implies \boxed{(30)_{10}}$

(3)

$\implies (75)_{10}$

Divide by 2,

⇒ Quotient 37, remainder 1.

Dividing the quotient by 2.

⇒ Quotient 18, remainder 1.

Again,

⇒ Quotient 9, remainder 0.

Keep repeating,

⇒ Quotient 4, remainder 1.

⇒ Quotient 2, remainder 0.

⇒ Quotient 1, remainder 0.

Let the quotient be the greatest digit, and list the remainders in ascending order.

$\implies \boxed{(1001011)_{2}}$

(4)

Using the same process,

⇒ Quotient 25, remainder 0.

Dividing the quotient by 3.

⇒ Quotient 8, remainder 1.

Again,

⇒ Quotient 2, remainder 2.

$\implies \boxed{(2210)_{3}}$

(5)

$\implies (132)_{5}$

Let's bring this number into base 10.

$= 5^{2}\cdot 1+5^{1}\cdot 3+5^{0}\cdot 2$

$=25+15+2$

$=(42)_{10}$

Using the repeated division by 4,

⇒ Quotient 10, remainder 2

⇒ Quotient 2, remainder 2

$=\boxed{(222)_{4}}$

More Information

• Why do we use repeated division?

The reason comes from the process of repeated division. Say, we are dividing 42 by 5 repeatedly.

⇒ Quotient 8, remainder 2

$\implies 42=5\cdot 8+2$

⇒ Quotient 1, remainder 3

$\implies 42=5\cdot(5\cdot 1+3)+2$

Further simplifying,

$\implies 42=5^{2}\cdot 1+5^{1}\cdot 3+5^{0}\cdot 2$

If we see it through, we can see all the place values are in terms of base 5. So, this is why we use the repeated division process for base conversion.