Question:
The sum of the numbers from 1 to 100, which are not divisible by 3 and 5.
The sum of the numbers from 1 to 100, which are not divisible by 3 and 5.
Show Hint
Find the sum 1 to 100, then take away the sum of multiples of 3 and the sum of multiples of 5.
Updated On: Jul 13, 2026
- 2946
- 2732
- 2632
- 2317
Show Solution
Verified By Collegedunia
The Correct Option is D
Solution and Explanation
Step 1: Find the sum of all numbers from 1 to 100.
The numbers 1 to 100 form an AP with first term 1, last term 100, and 100 terms. Using \(S=\frac{n}{2}(a+l)\):
\[ S = \frac{100}{2}(1+100) = 50\times101 = 5050 \]
Step 2: Find the sum of multiples of 3 up to 100.
The multiples of 3 up to 100 are \(3,6,9,\ldots,99\), an AP with common difference 3. The number of terms is \(\frac{99}{3}=33\).
\[ S_3 = \frac{33}{2}(3+99) = \frac{33}{2}\times102 = 33\times51 = 1683 \]
Step 3: Find the sum of multiples of 5 up to 100.
The multiples of 5 up to 100 are \(5,10,\ldots,100\), with \(\frac{100}{5}=20\) terms.
\[ S_5 = \frac{20}{2}(5+100) = 10\times105 = 1050 \]
Step 4: Remove the multiples of 3 and the multiples of 5 from the total.
The numbers that are not divisible by 3 and not divisible by 5 are found by taking every multiple of 3 and every multiple of 5 out of the total sum:
\[ \text{Required sum} = 5050 - 1683 - 1050 = 2317 \]
Options (A) 2946 and (B) 2732 do not match this computation, and option (C) 2632 comes from a different way of combining the two sets, which is not what this question is asking for.
Final Answer:
The required sum is 2317.
\[ \boxed{2317} \]
The numbers 1 to 100 form an AP with first term 1, last term 100, and 100 terms. Using \(S=\frac{n}{2}(a+l)\):
\[ S = \frac{100}{2}(1+100) = 50\times101 = 5050 \]
Step 2: Find the sum of multiples of 3 up to 100.
The multiples of 3 up to 100 are \(3,6,9,\ldots,99\), an AP with common difference 3. The number of terms is \(\frac{99}{3}=33\).
\[ S_3 = \frac{33}{2}(3+99) = \frac{33}{2}\times102 = 33\times51 = 1683 \]
Step 3: Find the sum of multiples of 5 up to 100.
The multiples of 5 up to 100 are \(5,10,\ldots,100\), with \(\frac{100}{5}=20\) terms.
\[ S_5 = \frac{20}{2}(5+100) = 10\times105 = 1050 \]
Step 4: Remove the multiples of 3 and the multiples of 5 from the total.
The numbers that are not divisible by 3 and not divisible by 5 are found by taking every multiple of 3 and every multiple of 5 out of the total sum:
\[ \text{Required sum} = 5050 - 1683 - 1050 = 2317 \]
Options (A) 2946 and (B) 2732 do not match this computation, and option (C) 2632 comes from a different way of combining the two sets, which is not what this question is asking for.
Final Answer:
The required sum is 2317.
\[ \boxed{2317} \]
Was this answer helpful?
0
0
Top XAT Quantitative Ability Questions
- Last year, Mr Basu bought two scooters. This year, he sold both of them for Rs 30,000 each. On one scooter, he earned a profit of 20%, and on the other, he made a loss of 20%. What was his net profit or loss?
- In an examination, the average marks obtained by students who passed was \(x\%\), while the average of those who failed was \(y\%\). The average marks of all the students taking the exam was \(z\%\). Find, in terms of \(x\), \(y\) and \(z\), the percentage of students taking the exam who failed.
- Three circles A, B and C have a common centre O. A is the inner circle, B is the middle circle, and C is the outer circle. P is a point on the outer circle C, and the radius OP cuts the inner circle at X and the middle circle at Y, such that \(OX = XY = YP\). The ratio of the area of the region between the inner and middle circles to the area of the region between the middle and outer circles is:
- The sides of a rhombus ABCD measure 2 cm each, and the difference between two of its angles is \(90^{\circ}\). Then the area of the rhombus is:
- If \(S_n\) denotes the sum of the first \(n\) terms of an Arithmetic Progression, and \(S_1 : S_4 = 1 : 10\), then the ratio of the first term to the fourth term is:
View More Questions
Top XAT Number System Questions
- How many numbers between 1 and 1000 (both excluded) are both squares and cubes?
- For \(n = 1, 2, 3, \ldots\), let \(T_n = 1^3 + 2^3 + \cdots + n^3\). Which one of the following statements is correct?
If \(n\) is a positive integer, let \(S(n)\) denote the sum of the positive divisors of \(n\), including \(n\) itself, and let \(G(n)\) be the greatest divisor of \(n\). If \(H(n) = \dfrac{G(n)}{S(n)}\), then which of the following is the largest?
Top XAT Questions
- Ashok, a maths teacher in the school, asked his students to guess his date of birth by giving the following hints.
Hint 1: He was born in 1995
Hint 2: The sum of the products of his birth date multiplied by 24 and his birthday month multiplied by 60 is 1284.
Find the sum of Ashok’s birth date, birth month, and birth year. - Solve for the number of integral values of \(x: \frac{(x^2-15x+47)(x-13)}{(x-8)}<0\).
- In an examination, Akhil scored 11.11% less than the pass marks and Akshay scored 12.5% more than the pass marks. Find the percentage of pass marks if the sum of the scores of Akhil and Akshay is 25% more than the total marks.
- The ratio of paint and oil in Tank A is 3: 4, whereas in Tank B, the respective ratio is 4: 5 . Then, 96 liters of a mixture consisting of paint and oil in the ratio of 5: 1 is added to Tank A, such that the respective ratio in Tank A becomes exactly the reverse of that in Tank B. Find the amount (in liters) of the mixture from Tank B that is poured into Tank A such that the amount of paint and water in Tank A become equal.
- The points (–4, 3) and (–1, –4) are two end points of a diagonal of a parallelogram. If the other diagonal has the equation 7y + 3x + c = 0, then find the value of c.
View More Questions