Question

Which is the least number that must be subtracted from $1856$, so that the remainder when divided by $7$, $12$, and $16$ is $4$?

• A. $137$
• B. $1361$
• C. $140$
• D. $172$

Hint:

When solving remainder problems with a constant remainder, subtract the remainder first, then find the LCM of the divisors.

Answer 1

The Correct Option is A

We are told that after subtracting a certain number from $1856$, the remainder when divided by $7$, $12$, and $16$ is $4$.
Let the number to be subtracted be $x$. Then:
\[ 1856 - x \equiv 4 \ (\text{mod } 7) \] \[ 1856 - x \equiv 4 \ (\text{mod } 12) \] \[ 1856 - x \equiv 4 \ (\text{mod } 16) \]
This means that $(1856 - x) - 4$ is divisible by $7$, $12$, and $16$.
Let $N = (1856 - x) - 4$. Then:
$N$ is divisible by LCM$(7, 12, 16)$.
First, LCM$(7, 12) = 84$, and LCM$(84, 16) = 336$.
So $N$ must be a multiple of $336$.
Thus: \[ 1856 - x - 4 = 336k \] \[ 1852 - x = 336k \] We want the least positive value of $x$, so choose the largest $k$ such that $336k<1852$.
$336 \times 5 = 1680$ (valid), and $336 \times 6 = 2016$ (too big).
So $N = 1680$, giving: \[ 1852 - x = 1680 \] \[ x = 1852 - 1680 = 172 \] Wait — but the question asks for the least number to be subtracted, so let's check smaller $k$. If $k = 4$, $N = 1344$, then $x = 1852 - 1344 = 508$ (larger than 172). So indeed the smallest subtraction possible is $172$.
This matches option (D), not (A). Correct answer is $172$.