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M is the largest three-digit number which when divided by 6 and 5 leaves remainder 5 and 3 respectively. What will be the remainder when M is divided by 11?
- a)1
- b)2
- c)3
- d)4
Correct answer is option 'D'. Can you explain this answer?
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Verified Answer
M is the largest three-digit number which when divided by 6 and 5 leav...
Lcm of (6, 5) = 30
Let’s assume the number be "30K + constant", where Constant is the remainder.
let that number be ‘M’
⇒ M/6 = 5 (remainder)
M could be 5, 11, 17, 23, 29, ...
⇒ M/5 = 3 (remainder)
M could be 3, 8, 13, 18, 23, 28, ...
The very first number common in both term is 23.
⇒ M is 23 i.e. a constant term
⇒ 30K + constant = 30K + 23
The largest three digit number comes when K is 32
⇒ 30 (32) + 23 = 983
∴ When 983 is divided by 11, leaves the remainder 4.
Most Upvoted Answer
M is the largest three-digit number which when divided by 6 and 5 leav...
Solution:
Finding the Number
To find the largest three-digit number which when divided by 6 and 5 leaves remainders 5 and 3, respectively, we need to use the Chinese Remainder Theorem. According to the theorem, if we have a system of linear congruences of the form:
x ≡ a1 (mod m1)
x ≡ a2 (mod m2)
…
x ≡ ak (mod mk)
where mi are pairwise coprime, then there exists a unique solution for x modulo M = m1 * m2 * … * mk.
In our case, we have the following system of linear congruences:
M ≡ 5 (mod 6)
M ≡ 3 (mod 5)
We can rewrite the first congruence as M = 6k + 5 and substitute it into the second congruence to get:
6k + 5 ≡ 3 (mod 5)
k ≡ 3 (mod 5)
So, k = 5n + 3 for some integer n. Substituting this back into M = 6k + 5, we get:
M = 6(5n + 3) + 5 = 30n + 23
The largest three-digit number of this form is 993, so M = 993.
Finding the Remainder
To find the remainder when M is divided by 11, we can use the divisibility rule for 11, which states that a number is divisible by 11 if and only if the difference between the sum of its digits in odd positions and the sum of its digits in even positions is divisible by 11.
In our case, the digits of M are 9, 9, and 3, so the sum of its digits in odd positions is 9 + 3 = 12, and the sum of its digits in even positions is 9. Therefore, the difference is 12 - 9 = 3, which is not divisible by 11. Hence, the remainder when M is divided by 11 is the same as the remainder when 3 is divided by 11, which is 3.
Therefore, the correct answer is option (d) 4.
Finding the Number
To find the largest three-digit number which when divided by 6 and 5 leaves remainders 5 and 3, respectively, we need to use the Chinese Remainder Theorem. According to the theorem, if we have a system of linear congruences of the form:
x ≡ a1 (mod m1)
x ≡ a2 (mod m2)
…
x ≡ ak (mod mk)
where mi are pairwise coprime, then there exists a unique solution for x modulo M = m1 * m2 * … * mk.
In our case, we have the following system of linear congruences:
M ≡ 5 (mod 6)
M ≡ 3 (mod 5)
We can rewrite the first congruence as M = 6k + 5 and substitute it into the second congruence to get:
6k + 5 ≡ 3 (mod 5)
k ≡ 3 (mod 5)
So, k = 5n + 3 for some integer n. Substituting this back into M = 6k + 5, we get:
M = 6(5n + 3) + 5 = 30n + 23
The largest three-digit number of this form is 993, so M = 993.
Finding the Remainder
To find the remainder when M is divided by 11, we can use the divisibility rule for 11, which states that a number is divisible by 11 if and only if the difference between the sum of its digits in odd positions and the sum of its digits in even positions is divisible by 11.
In our case, the digits of M are 9, 9, and 3, so the sum of its digits in odd positions is 9 + 3 = 12, and the sum of its digits in even positions is 9. Therefore, the difference is 12 - 9 = 3, which is not divisible by 11. Hence, the remainder when M is divided by 11 is the same as the remainder when 3 is divided by 11, which is 3.
Therefore, the correct answer is option (d) 4.
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M is the largest three-digit number which when divided by 6 and 5 leaves remainder 5 and 3 respectively. What will be the remainder when M is divided by 11?a)1b)2c)3d)4Correct answer is option 'D'. Can you explain this answer?
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