Math Snap

STEP 1

Assumptions1. We are performing operations in different bases (base five and base six).
. The operations include multiplication and addition.

STEP 2

For the first operation, we need to convert the numbers from base five to base ten. The number $31_{\text {five }}$ is converted as follows$$31_{\text {five }} = \times5^1 +1 \times5^0$$

STEP 3

Calculate the value of $31_{\text {five }}$ in base ten.
$$31_{\text {five }} =3 \times5 +1 \times1 =15 +1 =16$$

STEP 4

Similarly, convert $3_{\text {five }}$ from base five to base ten.
$$3_{\text {five }} =3 \times^0$$

STEP 5

Calculate the value of $3_{\text {five }}$ in base ten.
$$3_{\text {five }} =3 \times1 =3$$

STEP 6

Now, perform the multiplication operation in base ten.
$$16 \times3 =48$$

STEP 7

Convert the result back to base five. To do this, divide48 by5 and keep track of the remainders.
$$48 \div5 =9 \text{ remainder }3$$

STEP 8

Then divide the quotient by5 again and keep track of the remainder.
$$\div5 =1 \text{ remainder }4$$

STEP 9

Finally, divide the last quotient by5.
$$\div5 = \text{ remainder }$$

The remainders in reverse order give the result in base five.
$$48_{\text {ten }} =143_{\text {five }}$$So, $31_{\text {five }} \cdot3_{\text {five }} =143_{\text {five }}$.