Class 9 Exam > Class 9 Questions > Let N = 1421*1423*1425. What is the remainder...
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Let N = 1421*1423*1425. What is the remainder when N is divided by 12?
- a)0
- b)9
- c)3
- d)6
Correct answer is option 'C'. Can you explain this answer?
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Verified Answer
Let N = 1421*1423*1425. What is the remainder when N is divided by 12?...
1421 = 118*12 + 5 = 12k + 5, where k = 118.
1423 = 12k + 7
1425 = 12k + 9
N = (12k + 5)(12 k + 7)(12k + 9)
When N is divided by 12, the remainder is same as the remainder, when 5 * 7 * 9 = 315 is divided by 12.
5 * 7 * 9 = 35 * 9 = 315
315 = 312 + 3 = 12 * 26 + 3
The remainder is 3.
Most Upvoted Answer
Let N = 1421*1423*1425. What is the remainder when N is divided by 12?...
Solution:
We have to find the remainder when N is divided by 12.
Prime factorization of 12 = 2^2 * 3
Let's find the remainders when N is divided by 2^2 and 3 separately.
Remainder when N is divided by 2^2:
Since 1424 is divisible by 4, we only need to consider the remainders when 1421, 1423, and 1425 are divided by 4.
1421 ≡ 1 (mod 4)
1423 ≡ 3 (mod 4)
1425 ≡ 1 (mod 4)
Multiplying these three congruences, we get:
1421 * 1423 * 1425 ≡ 1 * 3 * 1 ≡ 3 (mod 4)
Therefore, the remainder when N is divided by 4 is 3.
Remainder when N is divided by 3:
Since the sum of the digits of N is equal to the sum of the remainders when the factors are divided by 3, we have:
N ≡ 1 + 4 + 2 + 1 + 1 + 4 + 2 + 3 + 1 + 4 + 2 + 5 ≡ 30 ≡ 0 (mod 3)
Therefore, the remainder when N is divided by 3 is 0.
Using the Chinese Remainder Theorem, we can combine the remainders when N is divided by 2^2 and 3 to find the remainder when N is divided by 12:
N ≡ 3 (mod 4)
N ≡ 0 (mod 3)
Solving this system of congruences, we get:
N ≡ 3 * 3 * 2 ≡ 18 ≡ 6 (mod 12)
Therefore, the remainder when N is divided by 12 is 6.
Hence, option 'C' is the correct answer.
We have to find the remainder when N is divided by 12.
Prime factorization of 12 = 2^2 * 3
Let's find the remainders when N is divided by 2^2 and 3 separately.
Remainder when N is divided by 2^2:
Since 1424 is divisible by 4, we only need to consider the remainders when 1421, 1423, and 1425 are divided by 4.
1421 ≡ 1 (mod 4)
1423 ≡ 3 (mod 4)
1425 ≡ 1 (mod 4)
Multiplying these three congruences, we get:
1421 * 1423 * 1425 ≡ 1 * 3 * 1 ≡ 3 (mod 4)
Therefore, the remainder when N is divided by 4 is 3.
Remainder when N is divided by 3:
Since the sum of the digits of N is equal to the sum of the remainders when the factors are divided by 3, we have:
N ≡ 1 + 4 + 2 + 1 + 1 + 4 + 2 + 3 + 1 + 4 + 2 + 5 ≡ 30 ≡ 0 (mod 3)
Therefore, the remainder when N is divided by 3 is 0.
Using the Chinese Remainder Theorem, we can combine the remainders when N is divided by 2^2 and 3 to find the remainder when N is divided by 12:
N ≡ 3 (mod 4)
N ≡ 0 (mod 3)
Solving this system of congruences, we get:
N ≡ 3 * 3 * 2 ≡ 18 ≡ 6 (mod 12)
Therefore, the remainder when N is divided by 12 is 6.
Hence, option 'C' is the correct answer.
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Community Answer
Let N = 1421*1423*1425. What is the remainder when N is divided by 12?...
3 is the right one
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Let N = 1421*1423*1425. What is the remainder when N is divided by 12?a)0b)9c)3d)6Correct answer is option 'C'. Can you explain this answer?
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