Question

What is the least number which when divided by 12, 18, 36, 45 leaves remainder 8, 14, 32, 41 respectively?

• A. 88
• B. 176
• C. 98
• D. 42

Hint:

In remainder problems, always check the difference between the divisor and the remainder first. If it's a constant value 'k', the answer is simply (LCM of divisors) - k.

Answer 1

The Correct Option is B

This is a classic problem involving LCM (Least Common Multiple). Step 1: Find the relationship between the divisors and their remainders. Let's calculate the difference between each divisor and its corresponding remainder.
Divisor 12, Remainder 8 \(\rightarrow\) Difference = 12 - 8 = 4
Divisor 18, Remainder 14 \(\rightarrow\) Difference = 18 - 14 = 4
Divisor 36, Remainder 32 \(\rightarrow\) Difference = 36 - 32 = 4
Divisor 45, Remainder 41 \(\rightarrow\) Difference = 45 - 41 = 4 The difference is the same in all cases. Let's call this common difference 'k', so k=4.

Step 2: Understand the formula for this type of problem. When the difference (Divisor - Remainder) is constant, the required number can be found using the formula: Required Number = (LCM of all divisors) - k

Step 3: Calculate the LCM of the divisors (12, 18, 36, 45). We can use prime factorization:
\(12 = 2 \times 2 \times 3 = 2^2 \times 3^1\)
\(18 = 2 \times 3 \times 3 = 2^1 \times 3^2\)
\(36 = 2 \times 2 \times 3 \times 3 = 2^2 \times 3^2\)
\(45 = 3 \times 3 \times 5 = 3^2 \times 5^1\) To find the LCM, we take the highest power of each prime factor present: \(2^2 \times 3^2 \times 5^1 = 4 \times 9 \times 5 = 180\). So, LCM(12, 18, 36, 45) = 180.

Step 4: Calculate the final number. Using the formula from Step 2: Required Number = LCM - k = 180 - 4 = 176.