Remainder. Find the greatest number that divides 48 and 76 with 6 as remainder in both cases
The Correct Answer and Explanation is:
Correct Answer:
The greatest number that divides 48 and 76, leaving a remainder of 6 in both cases, is 14.
Explanation
To solve this problem, we are given that the same number leaves a remainder of 6 when it divides both 48 and 76. Our goal is to find the greatest number that satisfies this condition.
Let us understand what this means mathematically.
If a number d divides 48 and leaves a remainder 6, then:
48 = d × q₁ + 6
⇒ 48 − 6 = d × q₁
⇒ 42 is divisible by d
Similarly, if the same number d divides 76 and leaves a remainder 6, then:
76 = d × q₂ + 6
⇒ 76 − 6 = d × q₂
⇒ 70 is divisible by d
This tells us that the number d divides both 42 and 70 exactly. Therefore, d must be a common divisor of 42 and 70.
Now, to find the greatest number that satisfies this condition, we must compute the greatest common divisor (GCD) of 42 and 70.
Let’s list the factors:
- Factors of 42: 1, 2, 3, 6, 7, 14, 21, 42
- Factors of 70: 1, 2, 5, 7, 10, 14, 35, 70
Common factors of 42 and 70 are: 1, 2, 7, 14
The greatest of these is 14.
Hence, the greatest number that divides both 48 and 76, leaving a remainder of 6 in each case, is 14.
Verification:
- 48 ÷ 14 = 3 remainder 6 ✅
- 76 ÷ 14 = 5 remainder 6 ✅
This confirms that 14 is the correct answer.
