To solve this problem, let's break it down into smaller steps and perform each calculation carefully. We are given:
- Divisor ([tex]\( d \)[/tex]) = 15
- Cociente ([tex]\( q \)[/tex]) = 36
- Residuo ([tex]\( r \)[/tex]) = 9
- Incremento al dividendo = 18
Firstly, we need to determine the original dividend ([tex]\( D \)[/tex]) before the increment. The relationship between dividend, divisor, quotient, and remainder is given by the formula:
[tex]\[ D = d \cdot q + r \][/tex]
Substituting the given values:
[tex]\[ D = 15 \cdot 36 + 9 \][/tex]
Calculating it gives us:
[tex]\[ D = 540 + 9 \][/tex]
[tex]\[ D = 549 \][/tex]
So, the original dividend is 549.
Next, we need to find the new dividend after increasing it by 18:
[tex]\[ D_{\text{new}} = D + 18 \][/tex]
[tex]\[ D_{\text{new}} = 549 + 18 \][/tex]
Calculating it gives us:
[tex]\[ D_{\text{new}} = 567 \][/tex]
Now, we determine the new quotient ([tex]\( q' \)[/tex]) and remainder ([tex]\( r' \)[/tex]) when the new dividend (567) is divided by the same divisor (15).
The relationship is still:
[tex]\[ D_{\text{new}} = d \cdot q' + r' \][/tex]
We need to calculate the integer division and the remainder:
[tex]\[ q' = \left\lfloor \frac{567}{15} \right\rfloor \][/tex]
[tex]\[ r' = 567 \mod 15 \][/tex]
Calculating [tex]\( q' \)[/tex]:
[tex]\[ q' = \left\lfloor \frac{567}{15} \right\rfloor \][/tex]
[tex]\[ q' = 37 \][/tex]
For the remainder [tex]\( r' \)[/tex]:
[tex]\[ r' = 567 \mod 15 \][/tex]
[tex]\[ r' = 12 \][/tex]
So, with the new dividend of 567, the new quotient is 37 and the new remainder is 12.
Thus, the new quotient after the increment is 37.
