I am a three-digit number
the sum of my digits is eighteen
Five letters are used to spell my second number
The difference between my first two digits is one.
I have fewer even digits than odd digits
My second digit is greater than my first digit
I am a multiple of seventy-five
If i add twenty-three my third digit is eight
Let's break this down step-by-step:
1. The sum of the digits is eighteen. This means that the three-digit number can be expressed as ABC, where A, B, and C are the digits. We know that A + B + C = 18.
2. Five letters are used to spell the second digit. This means that the second digit, B, has a spelled form consisting of five letters.
3. The difference between the first two digits is one. This implies that B - A = 1.
4. The number has fewer even digits than odd digits. This means that out of the three digits, there are more odd digits than even digits.
5. The second digit is greater than the first digit. So, B > A.
6. The number is a multiple of seventy-five. To find this number, we need to determine a multiple of 75 that satisfies the given conditions.
7. If we add twenty-three, the third digit becomes eight. This means that C + 23 = 8.
Now let's solve these conditions step-by-step:
From condition 3, we know that B - A = 1. There's only one pair of digits that differ by one: 8 and 9.
From condition 2, we know that the second digit, B, is spelled using five letters. The only number between 8 and 9 that fits this condition is 9 (nine).
From condition 4, we know that the number must have more odd digits than even digits. Since we have already used an odd digit (9), the remaining digits must be even. The only even digit remaining is 2.
Now let's check the other conditions with the values we have:
Condition 1: A + B + C = 18.
Plugging in the values A = 8, B = 9, we can solve for C: 8 + 9 + C = 18. Subtracting 17 from both sides, we get C = 1.
Condition 5: B > A.
We have B = 9 and A = 8, so B > A is true.
Condition 6: The number is a multiple of 75.
The number formed is 892, and we can check if it's divisible by 75. It turns out that 892 is not divisible by 75, so we need to try a different combination.
Since condition 6 is not satisfied, there is no valid three-digit number that satisfies all the given conditions.
To find the three-digit number that satisfies all the given conditions, we can use a systematic approach.
Condition 1: The sum of the digits is eighteen.
To find the digits that sum up to eighteen, we can start by testing different possibilities. Let's consider the following examples:
- 9 + 9 + 0 = 18
- 8 + 9 + 1 = 18
- 8 + 8 + 2 = 18
- 7 + 9 + 2 = 18
- 7 + 8 + 3 = 18
- 6 + 9 + 3 = 18
- 6 + 8 + 4 = 18
- 5 + 9 + 4 = 18
- 5 + 8 + 5 = 18
Condition 2: Five letters are used to spell the second digit.
Now we need to determine which combination of digits and letters produces a five-letter second digit. Let's consider the above examples again:
- 990: No five-letter combination for 9.
- 891: No five-letter combination for 8.
- 882: No five-letter combination for 8.
- 792: No five-letter combination for 9.
- 783: "Eight" consists of five letters.
- 693: No five-letter combination for 6.
- 684: No five-letter combination for 6.
- 594: No five-letter combination for 9.
- 585: No five-letter combination for 8.
Condition 3: The difference between the first two digits is one.
Based on the above possibilities, we can identify 783 as the only number that satisfies this condition.
Condition 4: Fewer even digits than odd digits.
In the number 783, we have one even digit (8) and two odd digits (7 and 3), which satisfies this condition.
Condition 5: The second digit is greater than the first digit.
In the number 783, the second digit (8) is indeed greater than the first digit (7), satisfying this condition.
Condition 6: The number is a multiple of seventy-five.
To check if 783 is a multiple of seventy-five, we can divide it by 75:
783 ÷ 75 = 10 Remainder 33
Since the remainder is not zero, 783 is not a multiple of seventy-five.
Condition 7: Adding twenty-three to the third digit gives us eight.
We can add twenty-three to the third digit of 783 to check if it equals eight:
3 + 23 = 26
Since the sum is not eight, 783 does not satisfy this condition.
From the given conditions, it seems that there is no three-digit number that satisfies all of them simultaneously.