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The greatest number which on dividing 1657 and 2037 leaves remainders 6 and 5 respectively, is:
- a)123
- b)127
- c)235
- d)305
Correct answer is option 'B'. Can you explain this answer?
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Most Upvoted Answer
The greatest number which on dividing 1657 and 2037 leaves remainders ...
Required number = H.C.F. of (1657 - 6) and (2037 - 5)
= H.C.F. of 1651 and 2032 = 127.
= H.C.F. of 1651 and 2032 = 127.
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Community Answer
The greatest number which on dividing 1657 and 2037 leaves remainders ...
Explanation:
Step 1: Find the LCM of (1657-6) and (2037-5)
To find the greatest number that divides 1657 and 2037 with remainders 6 and 5 respectively, we need to find the LCM of (1651) and (2032).
The LCM of two numbers is the smallest number that is divisible by both numbers.
Step 2: Calculate the LCM
Prime factorization of 1651:
1651 = 31 * 53
Prime factorization of 2032:
2032 = 2 * 2 * 2 * 127
Now, calculate the LCM by taking the highest power of each prime factor:
LCM = 2 * 2 * 2 * 31 * 53 * 127
LCM = 254012
Step 3: Subtract the remainders
Now, subtract the remainders from the LCM calculated:
254012 - 6 = 253006
254012 - 5 = 253007
Therefore, the greatest number which on dividing 1657 and 2037 leaves remainders 6 and 5 respectively is 253006.
Step 4: Check the options
Now, check the given options to find the closest number to 253006.
- 123: Not close to 253006
- 127: Closest to 253006
- 235: Not close to 253006
- 305: Not close to 253006
Hence, the correct answer is option B) 127.
Step 1: Find the LCM of (1657-6) and (2037-5)
To find the greatest number that divides 1657 and 2037 with remainders 6 and 5 respectively, we need to find the LCM of (1651) and (2032).
The LCM of two numbers is the smallest number that is divisible by both numbers.
Step 2: Calculate the LCM
Prime factorization of 1651:
1651 = 31 * 53
Prime factorization of 2032:
2032 = 2 * 2 * 2 * 127
Now, calculate the LCM by taking the highest power of each prime factor:
LCM = 2 * 2 * 2 * 31 * 53 * 127
LCM = 254012
Step 3: Subtract the remainders
Now, subtract the remainders from the LCM calculated:
254012 - 6 = 253006
254012 - 5 = 253007
Therefore, the greatest number which on dividing 1657 and 2037 leaves remainders 6 and 5 respectively is 253006.
Step 4: Check the options
Now, check the given options to find the closest number to 253006.
- 123: Not close to 253006
- 127: Closest to 253006
- 235: Not close to 253006
- 305: Not close to 253006
Hence, the correct answer is option B) 127.
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The greatest number which on dividing 1657 and 2037 leaves remainders 6 and 5 respectively, is:a)123b)127c)235d)305Correct answer is option 'B'. Can you explain this answer?
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