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- What is the least number which when divided by the number 3, 5, 6, 8, 10 and 12 leaves in each case a remainder 2 but when divided by 22 leaves no remainder ?
- Solution[By Examveda Team]
- Click here to read 1000+ Related Questions on Problems on H.C.F and L.C.M[Arithmetic Ability]
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- What is the least number which on being divided by 5 6 8 9 12 leaves in each case a remainder 1 but when divided by 13 leaves no remainder?
- What is the least number which on being divided by 5 10 12 and 15 leaves 2 as remainder in each case but if divided by 7 leaves no remainder?
- When 5 8 and 12 divide a multiple of 13 the remainders left are 3 6 and 10 respectively find the least such number?
- What is the least number when divided by 5 6 7 and 8 leaves a remainder 3?
A
312
B
962
C
1562
D
1586
Solution:
By option verification ,
962 is divided by 3,5,6,8,10 and 12 leaves remainder 2, but when divided by 13 leaves no remainder
[or]
L.C.M of 3,5,6,8,10,12 is 120
1×120+2=122 is not divisible by 13
2×120+2=242 is not divisible by 13
3×120+2=362 is not divisible by 13
4×120+2=482 is not divisible by 13
5×120+2=602 is not divisible by 13
6×120+2=722 is not divisible by 13
7×120+2=842 is not divisible by 13
8×120+2=962 is divisible by 13
What is the least number which when divided by the number 3, 5, 6, 8, 10 and 12 leaves in each case a remainder 2 but when divided by 22 leaves no remainder ?
A. 312
B. 242
C. 1562
D. 1586
Answer: Option B
Solution[By Examveda Team]
$$\eqalign{ & {\text{LCM of }}\left[ {3,5,6,8,10,12} \right] \cr & = 3 \times 5 \times 2 \times 4 \cr & = 120 \cr & {\text{Required number is }} \cr & \frac{{120K + 2}}{{22}} = \frac{{10K + 2}}{{22}} \cr & {\text{at }}K{\text{ = 2,}}\frac{{10K + 2}}{{22}} \cr & \Rightarrow {\text{Remainder = 0}} \cr & {\text{The given condition satisfied}} \cr & {\text{ = }}120K + 2 \cr & = 240 + 2 \cr & = 242 \cr} $$
Click here to read 1000+ Related Questions on Problems on H.C.F and L.C.M[Arithmetic Ability]
Given:
The least number which when divided by 5, 6, 7 and 8 leaves a remainder 3
The least number when divided by 9 remainder = 0
Concept Used:
For the least number take LCM and also check that all conditions are satisfying or not
For same remainder in each case add the remainder in the LCM
Calculation:
First we have to find the LCM of 5, 6, 7, 8
5 = 51
6 = 21 × 31
7 = 71
8 = 23
LCM of 5, 6, 7, 8 is 23 × 31 × 51 × 71 = 840
For same remainder we have to add 3 to the LCM
⇒ 840 + 3
Now check weather it is divisible by 9 or not
843/9 = Not divisible by 9
Now we have to take next multiple of LCM and then add 3
840 × 2 + 3
⇒ 1683
Divide it by 9
1683/9 = 187
⇒ 1683 is divisible by 9
⇒ 1683 is the least number when divided by 5, 6, 7, 8 gives remainder 3
Shortcut Trick
The least number must be divisible by 9
For divisibility of 9 sum of digits of the number must be divisible by 9
Checking it by options:
Option 1] 1677
1 + 6 + 7 + 7 = 21
⇒ 1677 is not divisible by 9
Option 2] 1683
1 + 6 + 8 + 3 = 18
⇒ 1683 is divisible by 9
Option 3] 2523
2 + 5 + 2 + 3 = 12
⇒ 2523 is not divisible by 9
Option 4] 3363
3 + 3 + 6 + 3 = 15
⇒ 3363 is not divisible by 9
∴ The least number is 1683 which when divided by 5, 6, 7 and 8 leaves a remainder 3, but when divided by 9 leaves no remainder.
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Updated On: 27-06-2022
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Text Solution
31224215621586
Answer : B
Answer
Step by step solution by experts to help you in doubt clearance & scoring excellent marks in exams.
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What is the least number which on being divided by 5 6 8 9 12 leaves in each case a remainder 1 but when divided by 13 leaves no remainder?
∴ The least number which is divided by 5, 6, 8, 9, 12 leaves a remainder 1 but when divided by 13 leaves no remainder is 3601. Option A is the correct option.
What is the least number which on being divided by 5 10 12 and 15 leaves 2 as remainder in each case but if divided by 7 leaves no remainder?
∴ The number is 64.
When 5 8 and 12 divide a multiple of 13 the remainders left are 3 6 and 10 respectively find the least such number?
Only one answer remains, 598.
What is the least number when divided by 5 6 7 and 8 leaves a remainder 3?
∴ The least number is 1683 which when divided by 5, 6, 7 and 8 leaves a remainder 3, but when divided by 9 leaves no remainder.