Find the greatest number which divides 120, 165 and 210 exactly leaving remainders 5, 4 and 3 respectively.

a) 7

b) 5

c) 23

d) None of these

To find the greatest number that divides 120, 165, and 210 exactly while leaving remainders of 5, 4, and 3 respectively, we can approach the problem step by step. The key is to first adjust each number by subtracting the respective remainders, which will help us find a common divisor.

Step 1: Adjust the Numbers

We start by subtracting the remainders from each number:

• For 120, subtract 5: 120 - 5 = 115
• For 165, subtract 4: 165 - 4 = 161
• For 210, subtract 3: 210 - 3 = 207

Step 2: Find the Greatest Common Divisor (GCD)

Next, we need to find the GCD of the adjusted numbers: 115, 161, and 207. The GCD is the largest number that can divide all three numbers without leaving a remainder.

Finding the GCD

We can use the prime factorization method or the Euclidean algorithm. Here, we'll use the prime factorization method:

• **115**: The prime factors are 5 and 23 (since 115 = 5 × 23).
• **161**: The prime factors are 7 and 23 (since 161 = 7 × 23).
• **207**: The prime factors are 3 and 69 (and 69 can be factored further into 3 × 23, so 207 = 3 × 69).

Now, let's identify the common prime factors:

• 115 has factors: 5, 23
• 161 has factors: 7, 23
• 207 has factors: 3, 23

The only common factor among all three numbers is 23.

Step 3: Verify the Solution

To ensure that 23 is indeed the greatest number that meets the criteria, we can check:

• 120 divided by 23 gives a remainder of 5: 120 = 23 × 5 + 5
• 165 divided by 23 gives a remainder of 4: 165 = 23 × 7 + 4
• 210 divided by 23 gives a remainder of 3: 210 = 23 × 9 + 3

Since all conditions are satisfied, we can confidently conclude that the greatest number that divides 120, 165, and 210 exactly while leaving the specified remainders is 23.

Final Answer

The correct option is c) 23.