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How many four-digit positive integers exist that contain the block 25 and are divisible by 75. (2250 and 2025 are two such numbers)?
- a)90
- b)63
- c)34
- d)87
- e)62
Correct answer is option 'C'. Can you explain this answer?
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Most Upvoted Answer
How many four-digit positive integers exist that contain the block 25 ...
Approach:
To find the four-digit positive integers that contain the block 25 and are divisible by 75, we need to consider the divisibility rules of 75. A number is divisible by 75 if it is divisible by both 3 and 25.
Divisibility by 3:
The sum of the digits of a number divisible by 3 is also divisible by 3. In this case, the sum of the digits (2 + 5 = 7) is not divisible by 3. Therefore, the number must be divisible by 3.
Divisibility by 25:
The last two digits of the number must be 25 for it to be divisible by 25.
Calculating the Range:
To calculate the number of four-digit positive integers that meet these criteria, we can consider the possible values for the first digit.
Calculating the First Digit:
Since the number is divisible by 3 and ends in 25, the possible values for the first digit are limited. The sum of the digits must be divisible by 3, so the first digit can only be 2, 5, or 8.
Calculating the Possibilities:
For each of these values, we can calculate the number of possibilities for the second digit. Then, we can calculate the total number of four-digit positive integers that meet the criteria.
Therefore, the correct answer is 34 four-digit positive integers that contain the block 25 and are divisible by 75.
To find the four-digit positive integers that contain the block 25 and are divisible by 75, we need to consider the divisibility rules of 75. A number is divisible by 75 if it is divisible by both 3 and 25.
Divisibility by 3:
The sum of the digits of a number divisible by 3 is also divisible by 3. In this case, the sum of the digits (2 + 5 = 7) is not divisible by 3. Therefore, the number must be divisible by 3.
Divisibility by 25:
The last two digits of the number must be 25 for it to be divisible by 25.
Calculating the Range:
To calculate the number of four-digit positive integers that meet these criteria, we can consider the possible values for the first digit.
Calculating the First Digit:
Since the number is divisible by 3 and ends in 25, the possible values for the first digit are limited. The sum of the digits must be divisible by 3, so the first digit can only be 2, 5, or 8.
Calculating the Possibilities:
For each of these values, we can calculate the number of possibilities for the second digit. Then, we can calculate the total number of four-digit positive integers that meet the criteria.
Therefore, the correct answer is 34 four-digit positive integers that contain the block 25 and are divisible by 75.
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Community Answer
How many four-digit positive integers exist that contain the block 25 ...
What will be the form of such 4-digit numbers?
The 4-digit numbers should contain the block 25.
The required 4-digit numbers will be of the form:
a. 25_ _
b. _ 25 _
c. _ _ 25
The required 4-digit numbers will be of the form:
a. 25_ _
b. _ 25 _
c. _ _ 25
What is the test of divisibility by 75?
If a number is divisible by 75, then it will be divisible by 25 and 3.
Count the number of 4-digit numbers for three possiblities
a. Numbers of the form 25 _ _ that are divisible by 75
A number of the form 25_ _ is divisible by 25 if its rightmost 2 digits are 00, 25, 50, or 75.
A number of the form 25_ _ is divisible by 25 if its rightmost 2 digits are 00, 25, 50, or 75.
Check which of these numbers are also multiples of 3
Only one number, 2550 satisfies the condition.
Only one number, 2550 satisfies the condition.
b. Numbers of the form _ 25 _ that are divisible by 75
A number of the form _ 25 _ is divisible by 25 if its unit digit is 0.
The 4-digit number will be of the form _ 250
A number of the form _ 25 _ is divisible by 25 if its unit digit is 0.
The 4-digit number will be of the form _ 250
What options exist for the left most digit so that the number is also divisible by 3?
The sum of the right most 3 digits of the number = 2 + 5 + 0 = 7.
If the first digit is 2 or 5 or 8, the sum of the 4 digits will be divisible by 3.
There are three 4-digit numbers that match the form _ 25 _ and are divisible by 75.
The sum of the right most 3 digits of the number = 2 + 5 + 0 = 7.
If the first digit is 2 or 5 or 8, the sum of the 4 digits will be divisible by 3.
There are three 4-digit numbers that match the form _ 25 _ and are divisible by 75.
c. Numbers of the form _ _ 25 that are divisible by 75
All numbesr of the form _ _ 25 is divisible by 25.
All numbesr of the form _ _ 25 is divisible by 25.
What options exist for the first 2 digits so that the number is also divisible by 3?
We already have a 2 and 5 whose sum is 7. 7 is a multiple of 3 plus 1.
We have a (3k + 1) with us. If we add a (3m + 2), the sum will be 3(k + m) + 1 + 2 = 3(k + m) + 3, which is divisible by 3.
The least 2 digit number that is of the form (3m + 2) is 11.
For example, if 11 takes the 1st 2 places, the number is divisible by 3
We already have a 2 and 5 whose sum is 7. 7 is a multiple of 3 plus 1.
We have a (3k + 1) with us. If we add a (3m + 2), the sum will be 3(k + m) + 1 + 2 = 3(k + m) + 3, which is divisible by 3.
The least 2 digit number that is of the form (3m + 2) is 11.
For example, if 11 takes the 1st 2 places, the number is divisible by 3
11 is not the only such number.
All 2-digit numbers of the form (3m + 2) will work
All 2-digit numbers of the form (3m + 2) will work
How many are there? The largest 2-digit number that is of the form 3m + 2 is 98.
And all of these numbers are in arithmetic progression with a common difference of 3
So, apply the arithmetic progression formula to compute the nth term: 98 = 11 + (n - 1)3
3(n - 1) = 87
(n - 1) = 29
Or n = 30
30 such 4-digit numbers exist
And all of these numbers are in arithmetic progression with a common difference of 3
So, apply the arithmetic progression formula to compute the nth term: 98 = 11 + (n - 1)3
3(n - 1) = 87
(n - 1) = 29
Or n = 30
30 such 4-digit numbers exist
Add the count of all three possiblities
1 + 3 + 30 = 34
34 such 4-digit numbers exist
34 such 4-digit numbers exist
Choice C is the correct answer.
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How many four-digit positive integers exist that contain the block 25 and are divisible by 75. (2250 and 2025 are two such numbers)?a)90b)63c)34d)87e)62Correct answer is option 'C'. Can you explain this answer?
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