Question

What is the highest number between 4000 and 5000, which when divided by 12, 16 and 24 would leave the remainder 4?

• A. 4699
• B. 6499
• C. 4969
• D. 4996

Answer 1

Answer:

D. 4996

We need a number (N) such that:

$N \equiv 4 \pmod{12}, \quad 16, \quad 24$

This means:
$N - 4 \text{ is divisible by } 12, 16, \text{ and } 24$
Find LCM of 12, 16, 24

• $(12 = 2^2 \times 3)$

• $(16 = 2^4)$

• $(24 = 2^3 \times 3)$

$\text{LCM} = 2^4 \times 3 = 48$

So,

$N = 48k + 4$

Find highest value between 4000 and 5000

$48k + 4 \le 5000$
$\Rightarrow 48k \le 4996$
$\Rightarrow k \le 104.08$

So (k = 104)

Calculate N

$N = 48 \times 104 + 4 = 4992 + 4 = 4996$

Check lower bound

4996 > 4000 \quad

Final Answer: 4996