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Two numbers were divided by a certain di...

Two numbers were divided by a certain divisor that left remainder 55 and 98. Now the sum of the numbers was divided by the same divisor and it left a remainder of 45. Find the divisor.

A

99

B

108

C

100

D

123

Text Solution

AI Generated Solution

The correct Answer is:
To solve the problem step by step, we will denote the divisor as \( D \), and the two numbers as \( N_1 \) and \( N_2 \). ### Step 1: Set up the equations based on the problem statement When \( N_1 \) is divided by \( D \), it leaves a remainder of 55. This can be expressed as: \[ N_1 = k_1 D + 55 \] for some integer \( k_1 \). When \( N_2 \) is divided by \( D \), it leaves a remainder of 98. This can be expressed as: \[ N_2 = k_2 D + 98 \] for some integer \( k_2 \). ### Step 2: Find the sum of the two numbers Now, let's find the sum \( N_1 + N_2 \): \[ N_1 + N_2 = (k_1 D + 55) + (k_2 D + 98) = (k_1 + k_2) D + (55 + 98) \] This simplifies to: \[ N_1 + N_2 = (k_1 + k_2) D + 153 \] ### Step 3: Analyze the remainder when the sum is divided by \( D \) According to the problem, when \( N_1 + N_2 \) is divided by \( D \), it leaves a remainder of 45. Therefore, we can express this as: \[ N_1 + N_2 \equiv 45 \mod D \] This means: \[ (k_1 + k_2) D + 153 \equiv 45 \mod D \] Since \( (k_1 + k_2) D \) is divisible by \( D \), we can ignore it in the modulo operation: \[ 153 \equiv 45 \mod D \] ### Step 4: Set up the equation from the modulo operation From the above equation, we can write: \[ 153 - 45 = nD \] for some integer \( n \). This simplifies to: \[ 108 = nD \] ### Step 5: Find the possible values for \( D \) To find the divisor \( D \), we need to find the divisors of 108. The divisors of 108 are: 1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54, 108. ### Step 6: Determine the valid divisor Since we know that the remainders (55 and 98) must be less than the divisor \( D \), we can eliminate any divisors that are less than or equal to 55. The possible divisors greater than 55 are: - 108 Thus, the only valid divisor \( D \) that satisfies all conditions is: \[ D = 108 \] ### Final Answer The divisor is \( D = 108 \).
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