find the greatest number which divides 225 and 2425 leaving remainder 5 in each case
The correct answer and explanation is:
Correct Answer: 20
Explanation:
To find the greatest number that divides both 225 and 2425 leaving a remainder of 5 in each case, start by understanding the concept involved.
If a number divides 225 and 2425 leaving a remainder of 5 in both cases, then subtract 5 from each of the numbers. This is because the actual number that divides both must divide the differences:
- 225 – 5 = 220
- 2425 – 5 = 2420
Now, the required number must exactly divide both 220 and 2420. That means the required number is a common divisor of 220 and 2420.
To find the greatest such number, calculate the Highest Common Factor (HCF) or Greatest Common Divisor (GCD) of 220 and 2420.
Step 1: Prime Factorization
220 = 2 × 2 × 5 × 11 = 2² × 5 × 11
2420 = 2 × 2 × 5 × 11 × 11 = 2² × 5 × 11²
Now find the common prime factors:
- Common prime factors: 2² × 5 × 11
So,
HCF = 2² × 5 × 11 = 4 × 5 × 11 = 220
But remember, we subtracted 5 earlier, so this 220 is the number that exactly divides both (225 – 5) and (2425 – 5). Hence, the greatest number that divides both 225 and 2425, leaving a remainder of 5, is:
Answer: 220
However, the question asks for the greatest number which divides 225 and 2425 leaving remainder 5. So the number must be less than both 225 and 2425, and divide the differences 220 and 2420. So now we double-check:
Find the HCF of 220 and 2420:
- 2420 ÷ 220 = 11
- So 220 is a divisor of 2420
Therefore, HCF = 220
Final Answer: 220
(Note: earlier I incorrectly wrote 20 in the final answer — correct value is 220)