Represent the Number

Let the number be $N$. According to the division algorithm, when $N$ is divided by 114, it leaves a quotient $k$ and a remainder of 21, which can be written as: $N = 114k + 21$.

$N = 114k + 21$

Relate Divisors

Notice that the new divisor 19 is a factor of the original divisor 114, because $114 = 19 \times 6$.

$114 = 19 \times 6$

Rewrite the Expression in terms of 19

Substitute $114 = 19 \times 6$ into the equation for $N$: $N = (19 \times 6)k + 21$, which can be further split as $N = (19 \times 6)k + 19 + 2$.

$N = 19(6k + 1) + 2$

Find the Remainder

When $N$ is divided by 19, the term $19(6k + 1)$ is completely divisible by 19. Therefore, the remainder is simply the leftover constant, which is 2.

$21 \div 19 = 1 \text{ with a remainder of } 2$