Let the number be represented as \( n \). According to the problem, when \( n \) is divided by 36, it leaves a remainder of 19. We can express this relationship mathematically as:
\[ n = 36k + 19 \]
for some integer \( k \).
Now, we want to find the remainder when \( n \) is divided by 12. To do this, we take the expression for \( n \) modulo 12:
\[ n \mod 12 = (36k + 19) \mod 12 \]
We will calculate \( 36k \mod 12 \) first. Since \( 36 \) is divisible by \( 12 \):
\[ 36k \mod 12 = 0 \quad \text{(for any integer } k\text{)} \]
Next, we find \( 19 \mod 12 \):
\[ 19 \div 12 = 1 \quad \text{(the integer part of the division)} \] \[ 19 - 12 \cdot 1 = 7 \] So,
\[ 19 \mod 12 = 7 \]
Now we can combine these results:
\[ n \mod 12 = (36k + 19) \mod 12 = (0 + 19) \mod 12 = 19 \mod 12 = 7 \]
Thus, the remainder when the number is divided by 12 is 7.
0 answers