Vincen wrote: ↑

Sat Jan 23, 2021 6:50 am

Positive integer \(A > 10,000\) and \(B > 20,000.\) When \(A\) is divided by \(237,\) the remainder is \(85.\) When \(B\) is divided by \(237,\) the remainder is \(41.\) What is the remainder when \(2A + 3B\) is divided by \(237?\)

a) 0
b) 56
c) 123
d) 170
e) 211

Answer: B

Source: Magoosh

It doesn’t matter what the values of \(A\) and \(B\) are. What matters is that we can add the remainders. Then,

\(\cdot\) If \(A\) leaves a remainder \(85,\) then \(2A\) will leave \(2\cdot 85=170.\)

\(\cdot\) If \(B\) leaves a remainder \(41,\) then \(3B\) will leave \(3\cdot 41=123.\)

So, \(2A+3B\) will leave \(170+123=293,\) but \(293=237+56.\) Therefore, the final remainder is \(56.\)

When A is divided by 237, the remainder is 85

Watch this video to learn the Basics of Remainders

Theory: Dividend = Divisor*Quotient + Remainder

A -> Dividend
237 -> Divisor
a -> Quotient (Assume)
85 -> Remainders
=> A = 237*a + 85 = 237a + 85

When B is divided by 237, the remainder is 41

Let quotient be b
=> B = 237*b + 41 = 237b + 41

What is the remainder when (2A + 3B) is divided by 237

2A + 3B = 2*(237a + 85) + 3*(237b + 41) = 237*2a + 170 + 237*3b + 123 = 237*(2a + 3b) + 237 + 56 = 237*(2a + 3b + 1) + 56

=> 2A + 3B when divided by 237 gives 2a + 3b + 1 as quotient and 56 as remainder.

So, Answer will be B
Hope it helps!

Watch this video to learn the Basics of Remainders