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In a two-digit, if it is known that its unit's digit exceeds its ten's digit by 2 and that the product of the given number and the sum of its digits is equal to 144, then the number is:
- a)24
- b)26
- c)42
- d)46
Correct answer is option 'A'. Can you explain this answer?
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Most Upvoted Answer
In a two-digit, if it is known that its units digit exceeds its tens d...
To solve this problem, let's assume the tens digit of the two-digit number as 'x' and the units digit as 'x+2'.
Let's break down the information given in the problem:
1. The units digit exceeds the tens digit by 2:
The units digit is 'x+2' and the tens digit is 'x'. Therefore, we can write the equation: (x+2) - x = 2. Simplifying, we get 2 = 2, which is true. This condition is satisfied.
2. The product of the number and the sum of its digits is equal to 144:
The given number is represented as 10x + (x+2), since it is a two-digit number. The sum of its digits is x + (x+2) = 2x + 2.
The product of the number (10x + (x+2)) and the sum of its digits (2x + 2) is equal to 144. So, we can write the equation: (10x + (x+2)) * (2x + 2) = 144.
Expanding the equation, we get: (11x + 2) * (2x + 2) = 144.
Simplifying further, we get: 22x^2 + 26x + 4 = 144.
Rearranging the equation, we have: 22x^2 + 26x - 140 = 0.
Now we need to factorize this quadratic equation. Dividing each term by 2, we get: 11x^2 + 13x - 70 = 0.
Factoring this equation, we find: (11x - 14)(x + 5) = 0.
Setting each factor to zero, we get two possible solutions:
11x - 14 = 0, which gives x = 14/11.
x + 5 = 0, which gives x = -5.
Since x represents the tens digit, it cannot be negative. Therefore, x = 14/11.
However, the tens digit must be a whole number, so we discard the solution x = 14/11.
Hence, the only possible solution is x = 2.
Therefore, the tens digit is 2 and the units digit is 4 (2+2).
So, the number is 24, which is option A.
Let's break down the information given in the problem:
1. The units digit exceeds the tens digit by 2:
The units digit is 'x+2' and the tens digit is 'x'. Therefore, we can write the equation: (x+2) - x = 2. Simplifying, we get 2 = 2, which is true. This condition is satisfied.
2. The product of the number and the sum of its digits is equal to 144:
The given number is represented as 10x + (x+2), since it is a two-digit number. The sum of its digits is x + (x+2) = 2x + 2.
The product of the number (10x + (x+2)) and the sum of its digits (2x + 2) is equal to 144. So, we can write the equation: (10x + (x+2)) * (2x + 2) = 144.
Expanding the equation, we get: (11x + 2) * (2x + 2) = 144.
Simplifying further, we get: 22x^2 + 26x + 4 = 144.
Rearranging the equation, we have: 22x^2 + 26x - 140 = 0.
Now we need to factorize this quadratic equation. Dividing each term by 2, we get: 11x^2 + 13x - 70 = 0.
Factoring this equation, we find: (11x - 14)(x + 5) = 0.
Setting each factor to zero, we get two possible solutions:
11x - 14 = 0, which gives x = 14/11.
x + 5 = 0, which gives x = -5.
Since x represents the tens digit, it cannot be negative. Therefore, x = 14/11.
However, the tens digit must be a whole number, so we discard the solution x = 14/11.
Hence, the only possible solution is x = 2.
Therefore, the tens digit is 2 and the units digit is 4 (2+2).
So, the number is 24, which is option A.
Community Answer
In a two-digit, if it is known that its units digit exceeds its tens d...
Let the ten's digit be x
Then, unit's digit = x + 2
Number = 10x + (x + 2) = 11x + 2
Sum of digits = x + (x + 2) = 2x + 2
∴ (11x + 2)(2x + 2) = 144
⇒ 22x2 + 26x - 140 = 0
⇒ 11x2 + 13x - 70 = 0
⇒ 11x2 + (35 - 22)x - 70 = 0
⇒ 11x2 + 35x - 22x - 70 = 0
⇒ (x - 2)(11x + 35) = 0
⇒ x = 2
Hence, required number = 11x + 2 = 24
Then, unit's digit = x + 2
Number = 10x + (x + 2) = 11x + 2
Sum of digits = x + (x + 2) = 2x + 2
∴ (11x + 2)(2x + 2) = 144
⇒ 22x2 + 26x - 140 = 0
⇒ 11x2 + 13x - 70 = 0
⇒ 11x2 + (35 - 22)x - 70 = 0
⇒ 11x2 + 35x - 22x - 70 = 0
⇒ (x - 2)(11x + 35) = 0
⇒ x = 2
Hence, required number = 11x + 2 = 24
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In a two-digit, if it is known that its units digit exceeds its tens digit by 2 and that the product of the given number and the sum of its digits is equal to 144, then the number is:a)24b)26c)42d)46Correct answer is option 'A'. Can you explain this answer?
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