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Certain 3-digit numbers following characteristics.
1. All the three digits are different.
2. The number is divisible by 7.
3. The number on reversing the digits is also divisible by 7.
How many such 3-digit numbers are there?
1. All the three digits are different.
2. The number is divisible by 7.
3. The number on reversing the digits is also divisible by 7.
How many such 3-digit numbers are there?
- a)2
- b)4
- c)6
- d)8
Correct answer is option 'B'. Can you explain this answer?
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Solution:
To find the required 3-digit numbers, we need to follow the given conditions:
Condition 1: All the three digits are different.
Condition 2: The number is divisible by 7.
Condition 3: The number on reversing the digits is also divisible by 7.
Let's start with Condition 2 first.
Condition 2:
The divisibility rule of 7 states that if we subtract twice the unit digit from the remaining tens digit, the resulting number must be divisible by 7. Let's use this rule to find the possible values of the tens digit and the unit digit.
Possible values of the tens digit:
Let's take the tens digit as x.
When we subtract twice the unit digit from the remaining tens digit, we get:
(10x + y) - 2z
where y and z are the ones and hundreds digits, respectively.
Since the number (10x + y) is divisible by 7, we can write:
10x + y = 7a
where a is some integer.
Now, substituting the value of (10x + y) in the above equation, we get:
7a - 2z = 10x + y - 2z
7a - 2z = 10x - 2(y - z)
Since 7a - 2z is divisible by 7, we can say that 10x - 2(y - z) must also be divisible by 7.
Let's list down the possible values of (y - z):
(y - z) = 1, 2, 3, 4, 5, 6, 7, 8
Substituting these values in the expression 10x - 2(y - z), we get:
10x - 2 = 8, 6, 4, 2, 0, -2, -4, -6
10x - 4 = 7, 5, 3, 1, -1, -3, -5, -7
10x - 6 = 6, 4, 2, 0, -2, -4, -6, -8
10x - 8 = 5, 3, 1, -1, -3, -5, -7, -9
10x - 10 = 4, 2, 0, -2, -4, -6, -8, -10
10x - 12 = 3, 1, -1, -3, -5, -7, -9, -11
10x - 14 = 2, 0, -2, -4, -6, -8, -10, -12
10x - 16 = 1, -1, -3, -5, -7, -9, -11, -13
Out of these, we can see that only two values of (y - z) give us a positive value of 10x - 2(y - z) that is divisible by 7:
(y - z) = 1, 8
Substituting these values in the expression 10x - 2(y - z), we get:
10x - 2 = 6
10x - 16 = -6
Hence,
To find the required 3-digit numbers, we need to follow the given conditions:
Condition 1: All the three digits are different.
Condition 2: The number is divisible by 7.
Condition 3: The number on reversing the digits is also divisible by 7.
Let's start with Condition 2 first.
Condition 2:
The divisibility rule of 7 states that if we subtract twice the unit digit from the remaining tens digit, the resulting number must be divisible by 7. Let's use this rule to find the possible values of the tens digit and the unit digit.
Possible values of the tens digit:
Let's take the tens digit as x.
When we subtract twice the unit digit from the remaining tens digit, we get:
(10x + y) - 2z
where y and z are the ones and hundreds digits, respectively.
Since the number (10x + y) is divisible by 7, we can write:
10x + y = 7a
where a is some integer.
Now, substituting the value of (10x + y) in the above equation, we get:
7a - 2z = 10x + y - 2z
7a - 2z = 10x - 2(y - z)
Since 7a - 2z is divisible by 7, we can say that 10x - 2(y - z) must also be divisible by 7.
Let's list down the possible values of (y - z):
(y - z) = 1, 2, 3, 4, 5, 6, 7, 8
Substituting these values in the expression 10x - 2(y - z), we get:
10x - 2 = 8, 6, 4, 2, 0, -2, -4, -6
10x - 4 = 7, 5, 3, 1, -1, -3, -5, -7
10x - 6 = 6, 4, 2, 0, -2, -4, -6, -8
10x - 8 = 5, 3, 1, -1, -3, -5, -7, -9
10x - 10 = 4, 2, 0, -2, -4, -6, -8, -10
10x - 12 = 3, 1, -1, -3, -5, -7, -9, -11
10x - 14 = 2, 0, -2, -4, -6, -8, -10, -12
10x - 16 = 1, -1, -3, -5, -7, -9, -11, -13
Out of these, we can see that only two values of (y - z) give us a positive value of 10x - 2(y - z) that is divisible by 7:
(y - z) = 1, 8
Substituting these values in the expression 10x - 2(y - z), we get:
10x - 2 = 6
10x - 16 = -6
Hence,
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Certain 3-digit numbers following characteristics.1. All the three digits are different.2. The number is divisible by 7.3. The number on reversing the digits is also divisible by 7.How many such 3-digit numbers are there?a)2b)4c)6d)8Correct answer is option 'B'. Can you explain this answer?
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