Question

When a number is divided by 36, it leaves a remainder of 19. What will be the remainder when the number is divided by 12?
19
7
10
192

Answer 1

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**.