Class 10 Exam > Class 10 Questions > The largest possible number with which when 3...
Start Learning for Free
The largest possible number with which when 38, 66
and 80 are divided, leaves the remainder same is
a)14
b)7
c)28
d)None of these
Correct answer is option 'A'. Can you explain this answer?
| FREE This question is part of | Download PDF Attempt this Test |
Verified Answer
The largest possible number with which when 38, 66... more and 80 are ...
Let the common remainder be 10
From the given numbers we now subtract the common remainder
i.e 38 − 10 = 28,66 −10 = 56, 80 −10 = 70
Now we will find the HCF of 28, 56, 70
28 = 2 x 7 x 2
56 = 2 x 2 x 2 x 7
70 = 2 x 5 x 7
So HCF of 28, 56 and 70 = 2 x 7
So the required number is 14
Most Upvoted Answer
The largest possible number with which when 38, 66... more and 80 are ...
Given: Numbers 38, 66, ..., 80
To find: The largest possible number with which when all these numbers are divided, leaves the same remainder.
Solution:
Step 1: Find the difference between consecutive numbers.
66 - 38 = 28
...
80 - 66 = 14
We see that the difference between consecutive numbers is not the same. Therefore, we cannot directly find the required number by taking the LCM of the given numbers.
Step 2: Let us assume that the required number is x.
Then, we can write:
38 ≡ r (mod x)
66 ≡ r (mod x)
...
80 ≡ r (mod x)
Here, 'r' is the common remainder.
Step 3: Subtracting the first two congruences, we get:
28 ≡ 0 (mod x)
Similarly, we can subtract the other congruences to get:
14 ≡ 0 (mod x)
...
80 - 66 ≡ 0 (mod x)
Step 4: Now, we need to find the largest possible number 'x' which divides all these differences (28, 14, ..., 14).
For this, we need to find the GCD of these differences.
GCD(28, 14, 14, 14) = 14
Therefore, the largest possible number 'x' is 14.
Step 5: Finally, we need to check if 14 satisfies the original congruences:
38 ≡ 4 (mod 14)
66 ≡ 12 (mod 14)
...
80 ≡ 8 (mod 14)
We see that 14 satisfies all the congruences. Therefore, the largest possible number with which all the given numbers can be divided, leaving the same remainder, is 14.
Hence, the correct answer is option (a) 14.
To find: The largest possible number with which when all these numbers are divided, leaves the same remainder.
Solution:
Step 1: Find the difference between consecutive numbers.
66 - 38 = 28
...
80 - 66 = 14
We see that the difference between consecutive numbers is not the same. Therefore, we cannot directly find the required number by taking the LCM of the given numbers.
Step 2: Let us assume that the required number is x.
Then, we can write:
38 ≡ r (mod x)
66 ≡ r (mod x)
...
80 ≡ r (mod x)
Here, 'r' is the common remainder.
Step 3: Subtracting the first two congruences, we get:
28 ≡ 0 (mod x)
Similarly, we can subtract the other congruences to get:
14 ≡ 0 (mod x)
...
80 - 66 ≡ 0 (mod x)
Step 4: Now, we need to find the largest possible number 'x' which divides all these differences (28, 14, ..., 14).
For this, we need to find the GCD of these differences.
GCD(28, 14, 14, 14) = 14
Therefore, the largest possible number 'x' is 14.
Step 5: Finally, we need to check if 14 satisfies the original congruences:
38 ≡ 4 (mod 14)
66 ≡ 12 (mod 14)
...
80 ≡ 8 (mod 14)
We see that 14 satisfies all the congruences. Therefore, the largest possible number with which all the given numbers can be divided, leaving the same remainder, is 14.
Hence, the correct answer is option (a) 14.
Free Test
FREE
| Start Free Test |
Community Answer
The largest possible number with which when 38, 66... more and 80 are ...
Assume that ‘d’ is the largest divisor , which divides 38 , 66 and 80 leaving remainder ‘ r' in each case. given no. = d*q + r , where 0 < r="" />< d="" dividend="divisor" *="" quotient+="" remainder="" 38-r="d" "="" *q1="66-r" ="d"="" *="" q2="" 80-r="d*" q3="" therefore.="" (38-r),(66-r)="&" (80-r)="" are="" exactly="" divisible="" by="" d="" as="" remainder="0." as="" we="" know="" that="" if="" d="" divides="" a="" &="" b.then="" d="" divides="" (a-="" b)="" also.="" so,="" d="" divides="" (66-r)="" -="" (38-r)="d" divides="" 28="" ………..(1)="" similarly="" d="" divides="" (80-="" r)="" -="" (38-r)="d" divides="" 42="" ……….(2)="" from="" (1)="" &="" (2)="" d="" is="" divides="" both="" 28,42="" 28="2*7*2" 42="2*7*3" so,="" hcf="" is="" 14="" ans:="" 14="" d="" dividend="divisor" *="" quotient="" +="" remainder="" 38="d" *="" q1="" +="" r="" 66="d" *="" q2="" +="" r="" 80="d" *="" q3="" +="" r="" or="" 38-r="d" *q1="" 66="" -r="d" *="" q2="" 80="" -r="d*" q3="" therefore.="" (38-r),="" (66-r)="" &="" (80-r)="" are="" exactly="" divisible="" by="" d="" as="" remainder="0." as="" we="" know="" that="" ,="" if="" d="" divides="" a="" &="" b.="" then="" d="" divides="" (a-="" b)="" also.="" so,="" d="" divides="" (66-r)="" -="" (38-r)="d" divides="" 28="" ………..(1)="" similarly="" d="" divides="" (80-="" r)="" -="" (38-r)="d" divides=" 42=" "="" ……….(2)="" from="" (1)="" &="" (2)="" d="" is="" divides="" both="" 28,42="" 28="2*7*2" 42="2*7*3" so,="" hcf="" is="" 14="" ans:="" d="" dividend="divisor" *="" quotient+="" remainder="" 38-r="d" "="" *q1="66-r" ="d"="" *="" q2="" 80-r="d*" q3="" therefore.="" (38-r),(66-r)="&" (80-r)="" are="" exactly="" divisible="" by="" d="" as="" remainder="0." as="" we="" know="" that="" if="" d="" divides="" a="" &="" b.then="" d="" divides="" (a-="" b)="" also.="" so,="" d="" divides="" (66-r)="" -="" (38-r)="d" divides="" 28="" ………..(1)="" similarly="" d="" divides="" (80-="" r)="" -="" (38-r)="d" divides="" 42="" ……….(2)="" from="" (1)="" &="" (2)="" d="" is="" divides="" both="" 28,42="" 28="2*7*2" 42="2*7*3" so,="" hcf="" is="" 14="" ans:="" 14="" d="" dividend="divisor" *="" quotient="" +="" remainder="" 38="d" *="" q1="" +="" r="" 66="d" *="" q2="" +="" r="" 80="d" *="" q3="" +="" r="" or="" 38-r="d" *q1="" 66="" -r="d" *="" q2="" 80="" -r="d*" q3="" therefore.="" (38-r),="" (66-r)="" &="" (80-r)="" are="" exactly="" divisible="" by="" d="" as="" remainder="0." as="" we="" know="" that="" ,="" if="" d="" divides="" a="" &="" b.="" then="" d="" divides="" (a-="" b)="" also.="" so,="" d="" divides="" (66-r)="" -="" (38-r)="d" divides="" 28="" ………..(1)="" similarly="" d="" divides="" (80-="" r)="" -="" (38-r)="d" divides=" 42=" "="" ……….(2)="" from="" (1)="" &="" (2)="" d="" is="" divides="" both="" 28,42="" 28="2*7*2" 42="2*7*3" so,="" hcf="" is="" 14="" ans:="" />
Attention Class 10 Students!
To make sure you are not studying endlessly, EduRev has designed Class 10 study material, with Structured Courses, Videos, & Test Series. Plus get personalized analysis, doubt solving and improvement plans to achieve a great score in Class 10.
|
Explore Courses for Class 10 exam
|
|
Similar Class 10 Doubts
Top Courses for Class 10View all
The largest possible number with which when 38, 66... more and 80 are divided, leaves the remainder same is a)14b)7c)28d)None of theseCorrect answer is option 'A'. Can you explain this answer?
Question Description
The largest possible number with which when 38, 66... more and 80 are divided, leaves the remainder same is a)14b)7c)28d)None of theseCorrect answer is option 'A'. Can you explain this answer? for Class 10 2024 is part of Class 10 preparation. The Question and answers have been prepared according to the Class 10 exam syllabus. Information about The largest possible number with which when 38, 66... more and 80 are divided, leaves the remainder same is a)14b)7c)28d)None of theseCorrect answer is option 'A'. Can you explain this answer? covers all topics & solutions for Class 10 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The largest possible number with which when 38, 66... more and 80 are divided, leaves the remainder same is a)14b)7c)28d)None of theseCorrect answer is option 'A'. Can you explain this answer?.
The largest possible number with which when 38, 66... more and 80 are divided, leaves the remainder same is a)14b)7c)28d)None of theseCorrect answer is option 'A'. Can you explain this answer? for Class 10 2024 is part of Class 10 preparation. The Question and answers have been prepared according to the Class 10 exam syllabus. Information about The largest possible number with which when 38, 66... more and 80 are divided, leaves the remainder same is a)14b)7c)28d)None of theseCorrect answer is option 'A'. Can you explain this answer? covers all topics & solutions for Class 10 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The largest possible number with which when 38, 66... more and 80 are divided, leaves the remainder same is a)14b)7c)28d)None of theseCorrect answer is option 'A'. Can you explain this answer?.
Solutions for The largest possible number with which when 38, 66... more and 80 are divided, leaves the remainder same is a)14b)7c)28d)None of theseCorrect answer is option 'A'. Can you explain this answer? in English & in Hindi are available as part of our courses for Class 10.
Download more important topics, notes, lectures and mock test series for Class 10 Exam by signing up for free.
Here you can find the meaning of The largest possible number with which when 38, 66... more and 80 are divided, leaves the remainder same is a)14b)7c)28d)None of theseCorrect answer is option 'A'. Can you explain this answer? defined & explained in the simplest way possible. Besides giving the explanation of
The largest possible number with which when 38, 66... more and 80 are divided, leaves the remainder same is a)14b)7c)28d)None of theseCorrect answer is option 'A'. Can you explain this answer?, a detailed solution for The largest possible number with which when 38, 66... more and 80 are divided, leaves the remainder same is a)14b)7c)28d)None of theseCorrect answer is option 'A'. Can you explain this answer? has been provided alongside types of The largest possible number with which when 38, 66... more and 80 are divided, leaves the remainder same is a)14b)7c)28d)None of theseCorrect answer is option 'A'. Can you explain this answer? theory, EduRev gives you an
ample number of questions to practice The largest possible number with which when 38, 66... more and 80 are divided, leaves the remainder same is a)14b)7c)28d)None of theseCorrect answer is option 'A'. Can you explain this answer? tests, examples and also practice Class 10 tests.
|
|
Explore Courses for Class 10 exam
|
|
Suggested Free Tests
Signup for Free!
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
|
© EduRev
|
Education Revolution
|
|
Signup to see your scores
go up within 7 days!
Access 1000+ FREE Docs, Videos and Tests
Takes less than 10 seconds to signup