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Find the remainder when 73 *75 *78 *57 *197 *37 is divided by 34.
- a)32
- b)10
- c)8
- d)Cannot be determined
Correct answer is option 'A'. Can you explain this answer?
Verified Answer
Find the remainder when 73 *75 *78 *57 *197 *37 is divided by 34.a)32b...
Remainder,
(73 *75 *78 *57 *197 *37)/34 ===> (5 *7 *10 *23 *27 *3)/34
[We have taken individual remainder, which means if 73 is divided by 34 individually, it will give remainder 5, 75 divided 34 gives remainder 7 and so on.]
(5 *7 *10 *23 *27 *3)/34 ===> (35 *30 *23 *27)/34 [Number Multiplied]
(35 *30 *23 *27)/34 ===> (1*-4*-11* -7)/34
[We have taken here negative as well as positive remainder at the same time. When 30 divided by 34 it will give either positive remainder 30 or negative remainder -4. We can use any one of negative or positive remainder at any time.]
(1 *-4 *-11 * -7)/34 ===> (28 *-11)/34 ===> (-6 *-11)/34 ===> 66/34 ===R===> 32.
Required remainder = 32.
Most Upvoted Answer
Find the remainder when 73 *75 *78 *57 *197 *37 is divided by 34.a)32b...
73*75*78*57*197*37÷34
firstly we will take common 2
73*75*39*57*197*37÷17
now we will apply remainder theorem
73/17- remainder (5)
75/17-remainder(7)
39/17-remainder(5)
57/17-remainder(6)
197/17-remainder(10)
37/17-remainder(3)
5*7*5*6*10*3/17
35*60*15/17
remainder will be 1,9,15
1*15*9/17
135/17
remainder(16)
16*2=32remainder
firstly we will take common 2
73*75*39*57*197*37÷17
now we will apply remainder theorem
73/17- remainder (5)
75/17-remainder(7)
39/17-remainder(5)
57/17-remainder(6)
197/17-remainder(10)
37/17-remainder(3)
5*7*5*6*10*3/17
35*60*15/17
remainder will be 1,9,15
1*15*9/17
135/17
remainder(16)
16*2=32remainder
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Community Answer
Find the remainder when 73 *75 *78 *57 *197 *37 is divided by 34.a)32b...
Solution:
We need to find the remainder when 73 * 75 * 78 * 57 * 197 * 37 is divided by 34.
Observation:
We can observe that 34 = 2 * 17. So, we can find the remainders when the given expression is divided by 2 and 17 separately using the Chinese Remainder Theorem.
Remainder when the expression is divided by 2:
When a number is divided by 2, the remainder is either 0 or 1 depending on whether the number is even or odd.
Let's check whether the given numbers are even or odd.
- 73 and 57 are odd.
- 75 and 197 are odd.
- 78 is even.
- 37 is odd.
The product of odd numbers is always odd.
So, the given expression is the product of 4 odd numbers and 2 even numbers.
- When we multiply an even number and an odd number, the product is even.
- When we multiply two odd numbers, the product is odd.
So, the product of 4 odd numbers and 2 even numbers is even.
Therefore, the remainder when the given expression is divided by 2 is 0.
Remainder when the expression is divided by 17:
Let's find the remainder when each number is divided by 17.
- 73 mod 17 = 5
- 75 mod 17 = 7
- 78 mod 17 = 10
- 57 mod 17 = 6
- 197 mod 17 = 11
- 37 mod 17 = 3
Using the properties of remainders under multiplication, we can write:
- (73 * 75 * 78 * 57 * 197 * 37) mod 17 = (5 * 7 * 10 * 6 * 11 * 3) mod 17
Now, we can find the product of these remainders and take the remainder when the product is divided by 17.
- (5 * 7 * 10 * 6 * 11 * 3) mod 17 = (2 * 7 * 4 * 11 * 3) mod 17
- = 1848 mod 17
- = 4
Therefore, the remainder when the given expression is divided by 17 is 4.
Using the Chinese Remainder Theorem, we can find the remainder when the expression is divided by 34.
- Let's find the values of M1 and M2 such that M1 * 2 + M2 * 17 = gcd(2, 17) = 1. We can use the Euclidean algorithm to find gcd(2, 17) = 1.
- 17 = 2 * 8 + 1
- 1 = 17 - 2 * 8
- So, M1 = -8 and M2 = 1.
- Using the values of M1 and M2, we can find the solution using the formula:
- x = a1 * M2 * N1 + a2 * M1 * N2
- where x is the solution, a1 and a2 are the remainders when the expression is divided by 2 and 17 respectively, N1 and N2 are the values of M2 and M1 modulo the corresponding modulus
We need to find the remainder when 73 * 75 * 78 * 57 * 197 * 37 is divided by 34.
Observation:
We can observe that 34 = 2 * 17. So, we can find the remainders when the given expression is divided by 2 and 17 separately using the Chinese Remainder Theorem.
Remainder when the expression is divided by 2:
When a number is divided by 2, the remainder is either 0 or 1 depending on whether the number is even or odd.
Let's check whether the given numbers are even or odd.
- 73 and 57 are odd.
- 75 and 197 are odd.
- 78 is even.
- 37 is odd.
The product of odd numbers is always odd.
So, the given expression is the product of 4 odd numbers and 2 even numbers.
- When we multiply an even number and an odd number, the product is even.
- When we multiply two odd numbers, the product is odd.
So, the product of 4 odd numbers and 2 even numbers is even.
Therefore, the remainder when the given expression is divided by 2 is 0.
Remainder when the expression is divided by 17:
Let's find the remainder when each number is divided by 17.
- 73 mod 17 = 5
- 75 mod 17 = 7
- 78 mod 17 = 10
- 57 mod 17 = 6
- 197 mod 17 = 11
- 37 mod 17 = 3
Using the properties of remainders under multiplication, we can write:
- (73 * 75 * 78 * 57 * 197 * 37) mod 17 = (5 * 7 * 10 * 6 * 11 * 3) mod 17
Now, we can find the product of these remainders and take the remainder when the product is divided by 17.
- (5 * 7 * 10 * 6 * 11 * 3) mod 17 = (2 * 7 * 4 * 11 * 3) mod 17
- = 1848 mod 17
- = 4
Therefore, the remainder when the given expression is divided by 17 is 4.
Using the Chinese Remainder Theorem, we can find the remainder when the expression is divided by 34.
- Let's find the values of M1 and M2 such that M1 * 2 + M2 * 17 = gcd(2, 17) = 1. We can use the Euclidean algorithm to find gcd(2, 17) = 1.
- 17 = 2 * 8 + 1
- 1 = 17 - 2 * 8
- So, M1 = -8 and M2 = 1.
- Using the values of M1 and M2, we can find the solution using the formula:
- x = a1 * M2 * N1 + a2 * M1 * N2
- where x is the solution, a1 and a2 are the remainders when the expression is divided by 2 and 17 respectively, N1 and N2 are the values of M2 and M1 modulo the corresponding modulus
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Find the remainder when 73 *75 *78 *57 *197 *37 is divided by 34.a)32b)10c)8d)Cannot be determinedCorrect answer is option 'A'. Can you explain this answer?
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