. (b) What are the quotient and remainder when -72 is divided by 14? Be sure to give your remainder as an integer
such that
.
(a) What are the quotient and remainder when 87 is divided by 12? Be sure to give your remainder as an integer
such that
. (b) What are the quotient and remainder when -72 is divided by 14? Be sure to give your remainder as an integer
such that
The Correct Answer and Explanation is:
Quotient and Remainder when 87 is divided by 12:
To find the quotient and remainder, we perform the division of 87 by 12.
- Division: 87÷12=7 (quotient)with a remainder.87 \div 12 = 7 \text{ (quotient)} \quad \text{with a remainder}.87÷12=7 (quotient)with a remainder.
- Calculation of remainder:
Multiply the quotient by the divisor: 7×12=847 \times 12 = 847×12=84 Now subtract 84 from 87: 87−84=387 – 84 = 387−84=3 So the remainder is 3.
Thus, the quotient is 7 and the remainder is 3.
(b) Quotient and Remainder when -72 is divided by 14:
Now, let’s perform the division of -72 by 14. To get the remainder as a non-negative integer, we follow the same steps but adjust for the negative dividend.
- Division: −72÷14=−5 (quotient)with a remainder.-72 \div 14 = -5 \text{ (quotient)} \quad \text{with a remainder}.−72÷14=−5 (quotient)with a remainder.
- Calculation of remainder:
Multiply the quotient by the divisor: −5×14=−70-5 \times 14 = -70−5×14=−70 Now subtract -70 from -72: −72−(−70)=−72+70=−2-72 – (-70) = -72 + 70 = -2−72−(−70)=−72+70=−2 To make the remainder a non-negative integer, we need to adjust it by adding the divisor (14): −2+14=12-2 + 14 = 12−2+14=12 Thus, the remainder becomes 12, and we adjust the quotient to reflect this change. The correct quotient is one less than -5 (i.e., -6).
Thus, the quotient is -6 and the remainder is 12.
Summary:
- (a) When 87 is divided by 12, the quotient is 7, and the remainder is 3.
- (b) When -72 is divided by 14, the quotient is -6, and the remainder is 12.
These calculations are consistent with the rule that the remainder must be a non-negative integer that is less than the divisor.
