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Q13:
How many three-digit numbers are divisible by 7?
Solution :
Given:
We are looking for three-digit numbers. The range of three-digit numbers is from $100$ to $999$. We are specifically interested in those numbers that are divisible by $7$.
To Find:
The total count of three-digit numbers divisible by $7$.
Step 1: Identifying the first three-digit number divisible by 7
The smallest three-digit number is $100$. Let us divide $100$ by $7$:
$100 \div 7 = 14$ with a remainder of $2$.
To find the next multiple of $7$, we calculate: $100 + (7 - 2) = 105$.
Thus, the first three-digit number divisible by $7$ is $a = 105$.
Step 2: Identifying the last three-digit number divisible by 7
The largest three-digit number is $999$. Let us divide $999$ by $7$:
$999 \div 7 = 142$ with a remainder of $5$.
To find the largest multiple of $7$ less than or equal to $999$, we subtract the remainder from $999$:
$999 - 5 = 994$.
Thus, the last three-digit number divisible by $7$ is $l = 994$.
Step 3: Formulating the Arithmetic Progression (AP)
The sequence of three-digit numbers divisible by $7$ forms an Arithmetic Progression where:
First term ($a$) = $105$
Common difference ($d$) = $7$
Last term ($a_n$) = $994$
Step 4: Applying the formula for the $n^{th}$ term of an AP
The formula for the $n^{th}$ term of an AP is given by:
$a_n = a + (n - 1)d$
[Where $a_n$ is the last term, $a$ is the first term, $d$ is the common difference, and $n$ is the number of terms.]
Step 5: Solving for $n$
Substitute the known values into the formula:
$994 = 105 + (n - 1)7$
Subtract $105$ from both sides:
$994 - 105 = (n - 1)7$
$889 = (n - 1)7$
Divide both sides by $7$:
$889 / 7 = n - 1$
$127 = n - 1$
Add $1$ to both sides:
$n = 127 + 1$
$n = 128$
Final Answer:
There are 128 three-digit numbers that are divisible by 7.
More Questions from Class 10 Mathematics Arithmetic Progression EXERCISE 5.2
- Q1(i): Fill in the blanks in the following table, given that $a$ is the first term, $d$ the common difference and $a_n$ the $n$th term of the AP: (i) $a=7, d=3, n=8, a_n=...$
- Q1(ii): Fill in the blanks in the following table, given that $a$ is the first term, $d$ the common difference and $a_n$ the $n$th term of the AP: (ii) $a=–18, d=..., n=10, a_n=0$
- Q1(iii): Fill in the blanks in the following table, given that $a$ is the first term, $d$ the common difference and $a_n$ the $n$th term of the AP: (iii) $a=..., d=–3, n=18, a_n=–5$
- Q1(iv): Fill in the blanks in the following table, given that $a$ is the first term, $d$ the common difference and $a_n$ the $n$th term of the AP: (iv) $a=–18.9, d=2.5, n=..., a_n=3.6$
- Q1(v): Fill in the blanks in the following table, given that $a$ is the first term, $d$ the common difference and $a_n$ the $n$th term of the AP: (v) $a=3.5, d=0, n=105, a_n=...$
- Q10: The 17th term of an AP exceeds its 10th term by 7. Find the common difference.
- Q11: Which term of the AP : 3, 15, 27, 39, . . . will be 132 more than its 54th term?
- Q12: Two APs have the same common difference. The difference between their 100th terms is 100, what is the difference between their 1000th terms?
- Q14: How many multiples of 4 lie between 10 and 250?
- Q15: For what value of $n$, are the $n$th terms of two APs: 63, 65, 67, . . . and 3, 10, 17, . . . equal?
- Q16: Determine the AP whose third term is 16 and the 7th term exceeds the 5th term by 12.
- Q17: Find the 20th term from the last term of the AP : 3, 8, 13, . . ., 253.
- Q18: The sum of the 4th and 8th terms of an AP is 24 and the sum of the 6th and 10th terms is 44. Find the first three terms of the AP.
- Q19: Subba Rao started work in 1995 at an annual salary of ₹ 5000 and received an increment of ₹ 200 each year. In which year did his income reach ₹ 7000?
- Q2(i): Choose the correct choice in the following and justify : (i) 30th term of the AP: 10, 7, 4, . . . , is
- Q2(ii): Choose the correct choice in the following and justify : (ii) 11th term of the AP: – 3, $-\frac{1}{2}$, 2, . . ., is
- Q20: Ramkali saved ₹ 5 in the first week of a year and then increased her weekly savings by ₹ 1.75. If in the $n$th week, her weekly savings become ₹ 20.75, find $n$.
- Q3(i): In the following APs, find the missing terms in the boxes : (i) 2, ☐, 26
- Q3(ii): In the following APs, find the missing terms in the boxes : (ii) ☐, 13, ☐, 3
- Q3(iii): In the following APs, find the missing terms in the boxes : (iii) 5, ☐, ☐, $9\frac{1}{2}$
- Q3(iv): In the following APs, find the missing terms in the boxes : (iv) – 4, ☐, ☐, ☐, ☐, 6
- Q3(v): In the following APs, find the missing terms in the boxes : (v) ☐, 38, ☐, ☐, ☐, – 22
- Q4: Which term of the AP : 3, 8, 13, 18, . . . ,is 78?
- Q5(i): Find the number of terms in each of the following APs : (i) 7, 13, 19, . . . , 205
- Q5(ii): Find the number of terms in each of the following APs : (ii) 18, $15\frac{1}{2}$, 13, . . . , – 47
- Q6: Check whether – 150 is a term of the AP : 11, 8, 5, 2 . . .
- Q7: Find the 31st term of an AP whose 11th term is 38 and the 16th term is 73.
- Q8: An AP consists of 50 terms of which 3rd term is 12 and the last term is 106. Find the 29th term.
- Q9: If the 3rd and the 9th terms of an AP are 4 and – 8 respectively, which term of this AP is zero?
CBSE Solutions for Class 10 Mathematics Arithmetic Progression
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