A sequence of positive integers starts with one 1, followed by two 2s, three 3s, and so on. (Each positive integer n occurs n times.) How many of the first 105 numbers in this sequence are divisible by 3?

Q: 3) Sarah writes down random positive integers when she gets bored. Prove that if Sarah writes 1001…

A: Given:  Sarah writes 1001 numbers.  Since there are 10 digits, i.e., there are only 103 = 1000…

Q: . How many 3-digit numbers can be formed from the digits 2, 3, 5, 6, 7 and 9, which are divisible by…

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Q: How many six-digit positive integers contain exactly one one, two twos and three treor? How many…

A: First note that here six digit integer can start only with 1 or 2. If it starts with 1 the remaining…

Q: Find S, the sum of first 15 Natural numbers 15

A: We have to find sum of first 15 Natural numbers.   S15? Solution :  As we know that the sum of first…

Q: Show that the sum of the cubes of the first n positive integers is (n(n + 1)/2)^2

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Q: In a set of 2 positive integers, how many ways it can be chosen in less than 50? 1.

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Q: How many natural numbers that are less or equal to 1000 and that are not multiples of 3, 5, or 7…

A: A number is said to be divisible by other number if on dividing the number  with it given remainder…

Q: How many positive integers less than 1,000,000 have a digit sum of 5?

A: We have to find the number of positive integers 1,000,000 have a digit sum of 5. The number less…

Q: Prove that the sum of any three consecutive numbers is divisible by 3

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Q: (a) Show that if seven integers are selected from the first 10 positive integers, there must be at…

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Q: 5. Prove that the sum of the first n natural numbers is equal to n(n+ 1)/2, and deduce that the sum…

A: Sum of n number is given by

Q: a) How many 3-digit positive integers are there with odd digits?

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Q: 2. In how many ways can three numbers be selected from the numbers 1, 2, ... , 300 such that their…

A: This problem is solved using the concept of permutation and combination. please find step by step…

Q: 1. How many positive integers of 5-digit-number can be formed from the prime numbers starting at 1…

A: Note: We’ll answer the first question since the exact one wasn’t specified. Please submit a new…

Q: How many numbers between 1 and 10000 have the digits 1,3, and 5 such that each digit appears exactly…

A: Here Numbers between 1 and 10,000 have digits 1,3,5 and each appears exactly once only

Q: Find the sum of the positive even integers up to and including 50.

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Q: 4. How many 5 digit numbers can be formed if a) the digits 2-8 are available and there must be at…

A: 4. It is required to form 5 digit numbers under two different sets of conditions.

Q: How many different integers must be selected to guarantee that at least 5 of them would have the…

A: Dividing any integer by 3, can have remainder 0, 1 or 2. Therefore, possibilities for remainder are…

Q: How many odd integers from 1,000 through 9,999 have distinct digits?

A: From 1000 to 9999 , all are 4 digited integers.  If the integer is odd then the last digit can occur…

Q: 6.) Find the sum of the first 50 positive integers ending in 6.

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Q: (b) Find the sum of the first 110 positive odd integers.

A: Our Aim is to find the sum of the first 110 positive odd integers.

Q: How many integers in the set {1,2,3,4, ..., 200} is divisible by at least one of the primes 2, 5 and…

A: Just use inclusion-exclusion principle for three sets.

Q: Find the sum of all four-digit numbers that can be formed by the digits 1, 3, 5, 7, 9 without…

A: Answer: Option b)  666600

Q: (a) Suppose that there are k distinct positive integers that divide an odd positive integer n.…

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Q: How many positive odd integers less than 10,000 can be written using the digits 3, 4, 6, 8, and 0?

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Q: How many positive odd integers less than 10,000 can be written using the digits 3, 4, 6, 8, and 0 if…

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Q: n how many distinct ways can the digits in the number 1,243,520,326 be arranged? (If there are any…

A: Given number 1,243,520,326 Total number of digits=10 Number of 1's=1 Number of 2's=3 Number of 3's=2…

Q: how many six-digit numbers can be formed so that 2 are in the place of thousandth and 5 in the place…

A: First number can not be 0. So we can pick the first number in 9 ways. (1 to 9)

Q: k is a prime number that has a ones digit of 7 and 0 <k < 100, how many values of k are possible?

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Q: how many positive integers n satisfy 22-5n>-3?

A: 22-5n&gt;-3

Q: a. On each of four consecutive plays in a football​ game, a team lost 8 yards. If lost yardage is…

A: On each of  four consecutive Plays in football gane team lost 8 yards

Q: How many integers between 100 and 400 contain the digit 2? There are integers between 100 and 400…

A: Numbers between 200 and 299 contains 2 So total integers = 100 Numbers between 120 – 129 contains 2…

Q: Find the average of the first 50 natural numbers is……………….

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Q: How many integers between 100 and 999 have distinct digits, and how many of those are even numbers?

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Q: How many integers from 1 to 99,999 contain exactly two consecutive 1s?

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Q: How many can 3 digits be formed using the digits from 1 to 5 if the digit 2 is never there in the…

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Q: For each natural number n, if 3 does not divide (n² + 2), then n is not a prime number or n = 3.

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Q: find the sum of all integers between 300 and 800 that are exactly divisible by 14

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Q: How many sequences of 6 positive digits (1-9) contain at least one digit that appears twice?

A: We have to determine the total sequences of 6 positive digits (1-9) that contain at least one digit…

Q: How large a subset of the integers from 1 to 50 must be chosen to guarantee that one of the numbers…

A: We have to determine How large a subset of integers from 1 to 50 must be chosen to guarantee that…

Q: Find the sum of all integers from 41 to 83. The sum of all integers from 41 to 83 is

A: Detailed Solution and Explanation is given below....

Q: How many sequences of 6 positive digits (1-9) contain at most one even digits?

A: Given number of digits in each sequence=6. Number of digits which can be used=1-9 Number of even…

Q: Find the sum of all natural numbers n such that 2021-n/99 is a natural number ?

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Q: Two numbers a and b are chosen from the set 1,2,.., 3n In how many ways can these integers be…

A: In the question we have to find total no. of cases by which a2-b2 is divisible by 3.

Q: The product of all the natural numbers from 1 to n is written as n! it is read as "n factorial" true…

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Q: Prove that the sum of two consecutive odd integers is divisible by 4.

A: Let the two consecutive integers be n and n+2 Then we need to prove that (n)+(n+2) is divisible by 4…

Q: How many corresponding integers are made up of 1443 digits taken from (1-2-3-4-5-6) provided that no…

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Q: How many four-digit positive integers both start and end in even digits:

A: We need to find the number of positive integers both start and end in even digits.

Q: A man finished a job in 5 days. On the first day, he finished 1/m of the job. On the second day, he…

A: GIVEN THAT JOB FINISHED = 5 DAYS