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A certain number when successively divided by 4, 5 and 7 leaves remainders 2, 3 and 5 respectively. Find such a least number.
- a)190
- b)112
- c)156
- d)138
Correct answer is option 'D'. Can you explain this answer?
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Verified Answer
A certain number when successively divided by 4, 5 and 7 leaves remain...
See Diference between Divisor and remaider
4-2=2 = k
5-3=2 = k
7-5=2 = k
this difference is same
so answer will be LCM(4,5,7) - k
140 - 2
138
4-2=2 = k
5-3=2 = k
7-5=2 = k
this difference is same
so answer will be LCM(4,5,7) - k
140 - 2
138
Most Upvoted Answer
A certain number when successively divided by 4, 5 and 7 leaves remain...
U can do this by hit and trial
divide each option by 9,7,5 and when u will get 8 ,5,1 as remainder that will be ur answer
divide each option by 9,7,5 and when u will get 8 ,5,1 as remainder that will be ur answer
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Community Answer
A certain number when successively divided by 4, 5 and 7 leaves remain...
Given: Number when divided by 9, 7, and 5 leaves remainders 8, 5, and 1 respectively.
To find: Smallest 4-digit number satisfying the given conditions.
Solution:
Let the number be x.
x ≡ 8 (mod 9)
x ≡ 5 (mod 7)
x ≡ 1 (mod 5)
We can use the Chinese Remainder Theorem (CRT) to solve the above system of congruences.
Step 1: Find the product of the moduli of the given congruences.
M = 9 x 7 x 5 = 315
Step 2: Find the values of Mi such that Mi ≡ M/mi (mod mi), where mi is the modulus of each congruence.
M1 = 315/9 = 35
M2 = 315/7 = 45
M3 = 315/5 = 63
Step 3: Find the solutions of the given congruences with respect to each modulus using the extended Euclidean algorithm.
x1 ≡ 1 (mod 9) => x1 = 8 + 9k1, where k1 is an integer.
x2 ≡ 1 (mod 7) => x2 = 5 + 7k2, where k2 is an integer.
x3 ≡ 1 (mod 5) => x3 = 1 + 5k3, where k3 is an integer.
Step 4: Find the solution of the system of congruences using CRT.
x ≡ Σ(xiMiMi^-1) (mod M), where xi = MiMi^-1 (mod mi) and Mi^-1 is the inverse of Mi modulo mi.
x ≡ (8 x 45 x 4) + (5 x 35 x 2) + (1 x 63 x 1) (mod 315)
x ≡ 1440 (mod 315)
The smallest 4-digit number satisfying the given conditions is 1061, which is the smallest number greater than or equal to 1440 that leaves a remainder of 8 when divided by 9, a remainder of 5 when divided by 7, and a remainder of 1 when divided by 5.
Therefore, the correct answer is option (D) 1061.
To find: Smallest 4-digit number satisfying the given conditions.
Solution:
Let the number be x.
x ≡ 8 (mod 9)
x ≡ 5 (mod 7)
x ≡ 1 (mod 5)
We can use the Chinese Remainder Theorem (CRT) to solve the above system of congruences.
Step 1: Find the product of the moduli of the given congruences.
M = 9 x 7 x 5 = 315
Step 2: Find the values of Mi such that Mi ≡ M/mi (mod mi), where mi is the modulus of each congruence.
M1 = 315/9 = 35
M2 = 315/7 = 45
M3 = 315/5 = 63
Step 3: Find the solutions of the given congruences with respect to each modulus using the extended Euclidean algorithm.
x1 ≡ 1 (mod 9) => x1 = 8 + 9k1, where k1 is an integer.
x2 ≡ 1 (mod 7) => x2 = 5 + 7k2, where k2 is an integer.
x3 ≡ 1 (mod 5) => x3 = 1 + 5k3, where k3 is an integer.
Step 4: Find the solution of the system of congruences using CRT.
x ≡ Σ(xiMiMi^-1) (mod M), where xi = MiMi^-1 (mod mi) and Mi^-1 is the inverse of Mi modulo mi.
x ≡ (8 x 45 x 4) + (5 x 35 x 2) + (1 x 63 x 1) (mod 315)
x ≡ 1440 (mod 315)
The smallest 4-digit number satisfying the given conditions is 1061, which is the smallest number greater than or equal to 1440 that leaves a remainder of 8 when divided by 9, a remainder of 5 when divided by 7, and a remainder of 1 when divided by 5.
Therefore, the correct answer is option (D) 1061.
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