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This Lesson (OVERVIEW of miscellanious solved problems on divisibility of integer numbers) was created by by ikleyn(52908)
  About ikleyn: OVERVIEW of miscellaneous solved problems on divisibility of integer numbers- Light flashes on a Christmas tree and a Least Common Multiple - The number that leaves a remainder 1 when divided by 2, by 3, by 4, by 5 and so on until 9 - The number which gives remainder 4 when divided by 7, remainder 5 when divided by 8 and remainder 6 when divided by 9 - Introductory problems on divisibility of integer numbers - Finding Greatest Common Divisor of integer numbers - Relatively prime numbers help to solve problems - Solving equations in integer numbers - Quadratic polynomial with odd integer coefficients can not have a rational root - Proving an equation has no integer solutions - Composite number of the form (4n+3) must have a prime divisor of the form (4n+3) - Problems on divisors of a given number - How many three-digit numbers are multiples of both 5 and 7? - How many 3-digit numbers are not divisible by 2; not divisible by 3; not divisible by either 2 or 3 - How many integer numbers in the range 1-300 are divisible by at least one of the integers 4, 6 and 15 ? - Find the remainder of division - Why 3^n + 7^n - 2 is divisible by 8 for all positive integer n ? - What is the last digit of the number a^n ? - Find the last three digits of these numbers - Find the last two digits of the number 3^123 + 7^123 + 9^123 - Find the last two digits of (1! + 2! + 3! + ... + 2024!)^2024 - Find n-th term of a sequence - Solving Diophantine equations - How many integers of the form n^2 + 18n + 13 are perfect squares - Miscellaneous problems on divisibility numbers - Find the sum of digits of integer numbers - Two-digit numbers with digit "9" - Find a triangle with integer side lengths and integer area - Math circle level problem on the hundred-handed monster Briareus - Math Circle level problem on lockers and divisors of integer numbers - Nice entertainment problems related to divisibility properties - Solving problems on modular arithmetic - Using the little Fermat's theorem to solve a problem on modular arithmetic List of lessons with short annotationsLight flashes on a Christmas tree and a Least Common MultipleProblem 1. On a Christmas tree, a first set of lights flashes every 3 seconds, a second set of lights flashes every 4 seconds, and a third set of lights flashes every 10 seconds. If the lights all came on at 10:00 am, at what time will they all come on together again? Problem 2. School starts at 8:00 AM. A bell rings every 60 minutes, and another bell rings every 48 minutes. If both bells ring at the start of the day, how many minutes will it be before they ring at the same time again? Problem 3. Two truckers leave Miami at the same time. They take 14 and 6 days, respectively, to reach their destination and return to Miami. The truckers each take continuous trips to and from Miami. How many days will pass before the two truckers leave Miami on the same day again? Problem 4. Three friends are packing sweets into gift boxes. They agree that each box should contain the same number of sweets, but they are each working in separate locations with their own pile of sweets so cannot share boxes. Gwen has 286 sweets, Bill has 390 sweets and Ariel has 468 sweets. If they put the largest number of sweets into each box that they can and they use up all their sweets, how many boxes of sweets will they pack? The number that leaves a remainder 1 when divided by 2, by 3, by 4, by 5 and so on until 9 Problem 1. Find the number less than 3000 that leaves a remainder 1 when divided by 2, by 3, by 4, by 5 and so on until 9. Problem 2. Find the number less than 3000 that leaves a remainder 1 when divided by 2, a remainder 2 when divided by 3, a remainder 3 when divided by by 4, a remainder 4 when divided by 5 and so on until 9. Problem 3. Find the greatest 5-digit number which when divided by 25, 30, 40 leaves a remainder of 20, 25 and 35 respectively. Problem 4. A band master wished to arrange the band members in pairs for a marching patterns. He found that he was one person short. He tried to arrange by 5s and 7s and he was still one person short. What is the least number of people in the marching band? Problem 5. In yoga class, all of the students are lined up according to height. Andy notices that the number of students who are taller than he is one-fourth the number of students who are shorter than he. Violetta notices that there are 3 times as many students who are taller than she than students who are shorter than she. How many students are the class if there are fewer than 40 students? The number which gives remainder 4 when divided by 7, remainder 5 when divided by 8 and remainder 6 when divided by 9 Problem 1. Find the least positive integer number satisfying each of the following conditions : - Divided by 7 gives a remainder of 4. - Divided by 8 gives a remainder of 5. - Divided by 9 gives a remainder of 6. Introductory problems on divisibility of integer numbers Problem 1. For any integer n, prove that 3 divides one of the integers n, n + 1 or 2n + 1. Problem 2. For any integer n, prove that 3 divides one of n, n + 2 or n + 4. Problem 3. For any integer n, prove that 3 divides one of n, 2n - 1 or 2n +1. Problem 4. Show that if n is an odd integer, then Problem 5. Prove that for any integer n, 5 divides Problem 6. Prove that Problem 7. How many positive integers " n " are there such that 2n+1 is a divisor of 8n+46 ? Problem 8. A six-digit number is formed by repeating a three-digit number as in 639639. Find the sum of the prime factors all such numbers have in common. Finding Greatest Common Divisor of integer numbers Problem 1. n is a natural number. Find the common Greatest Common Divisor (GCD) of numbers (2n+25) and (n+15). Problem 2. n is a natural number. Find the Greatest Common Divisor (GCD) of numbers 2(n+8) and (n+13). Problem 3. A rectangular box measuring 150 cm by 156 cm by 216 cm is to be packed with exact number of cubes of maximum lengths. Calculate then volume of one such cube Relatively prime numbers help to solve problems Problem 1. A group of boys and girls sit a test. Exactly 2/3 of the boys and 3/4 of the girls pass the test. If an equal number of boys and girls passed the test, what fraction of the entire group passed the test? Problem 2. The numerator of a fraction is increased by 8 and the denominator is decreased by 1, the resulting fraction is the reciprocal of the original fraction. What is the original fraction? Solving equations in integer numbers Problem 1. Solve equation in integer numbers Problem 2. Solve equation in integer numbers Problem 3. Find the solution in positive integer numbers x, y to equation Problem 4. Solve equation Quadratic polynomial with odd integer coefficients can not have a rational root Problem 1. Given odd integers a, b, c, prove that the equation Proving an equation has no integer solutions Problem 1. Prove that the equation Composite number of the form (4n+3) must have a prime divisor of the form (4n+3) Problem 1. Prove that any composite number of the form (4n+3) must have at least one prime factor of the form (4n+3). Problems on divisors of a given number Problem 1. Find the greatest 4 digit number that has exactly 3 factors. Problem 2. What is the smallest positive integer that has exactly 6 divisors ? Problem 3. Find all numbers between 200 and 500 which have exactly 9 factors. Problem 4. The numbers 1998, 2997, 3996, 4995, ..., 8991 all have two distinct primes in common in their prime factorization. Find the sum of these two primes. How many three-digit numbers are multiples of both 5 and 7? Problem 1. If X is the set of multiples of 5, and Y is the set of three-digit positive integers which are multiples of 7, how many numbers are common to both sets ? How many 3-digit numbers are not divisible by 2; not divisible by 3; not divisible by either 2 or 3 Problem 1. How many 3-digit numbers are - Not divisible by 2 ? - Not divisible by 3 ? - Not divisible by either 2 or 3 ? How many integer numbers in the range 1-300 are divisible by at least one of the integers 4, 6 and 15 ? Problem 1. How many integer numbers in the range 1-300 are divisible by at least one of the integers 4, 6 and 15 ? Problem 2. Find the number of positive integers between 1 and 600 inclusive that are not divisible by 4 or 5 or 6. Find the remainder of division Problem 1. Find the remainder modulo 36 of Problem 2. What remainder is left when the number Problem 3. What remainder is left when the number Problem 4. Find the remainder when Why 3^n + 7^n - 2 is divisible by 8 for all positive integer n ? Problem 1. Show that Problem 2. Prove that What is the last digit of the number a^n ? Problem 1. What is the last digit of the number Problem 2. What is the units digit of the number Problem 3. Determine the ONES digit of the number Problem 4. Find the last digit of the sum Find the last three digits of these numbers Problem 1. Using binomial theorem, find the Last three digits of the number Problem 2. Find the last three digits of the number Find the last two digits of the number 3^123 + 7^123 + 9^123 Problem 1. Find the last two digits in the decimal numeral for Problem 2. Find the last two digits of the number 3^123 + 7^123 + 9^123. Find the last two digits of (1! + 2! + 3! + ... + 2024!)^2024 Problem 1. Find the last two digits of Find n-th term of a sequence Problem 1. The increasing sequence {2, 3, 5, 6, 7, 8, 10, 11, . . .} consists of all positive integers that are not perfect squares. What is the 2012-th term of the sequence? Solving Diophantine equations Problem 1. Find the ordered pair (m,n), where m,n are positive integers satisfying the following equation 14mn = 55 - 7m - 2n. Problem 2. If x = 100 and y = 1, then 19x + 83y = 1983. There is exactly one other pair of positive integers x and y, for which this equation is true. What is this pair of integers? Problem 3. In a block of flats there are 24 units of 3 types: the luxury unit, the superior unit and deluxe unit. The luxury can accommodate 8 people, the superior unit can accommodate 7 people and deluxe can accommodate 5 people. Given that the total number of people living in this block is 160, how many of each type are there? Problem 4. Alan has 13 pieces of $2, $5 and $10 notes in his wallet. If the amount of money he has is $67, how many pieces of each note does he have ? Problem 5. Solve equation Problem 6. If 4x + 4xy + 4y = 260, where x and y are integer numbers, find all possible values of x+y. Problem 7. If Problem 8. For this given system of equations in integer numbers a + b + c = 6, find the value of How many integers of the form n^2 + 18n + 13 are perfect squares Problem 1. How many integers of the form n^2 + 18n + 13 are perfect squares? Problem 2. Prove that there are infinitely many natural numbers n such that Miscellaneous problems on divisibility numbers Problem 1. What is the greatest integer, which when divided into 383, 527, or 815 leaves the same remainder ? Problem 2. Given the sequense 2, 5, 8, . . . , prove that none of the terms is a perfect square. Problem 3. For any four positive integers a < b < c < d, six pairs of integers may be formed with the positive difference between integers in each pair. Prove that the product of these six positive differences is divisible by 12. Problem 4. The positive integers 1, 2, 3, 4, . . . , m are written one after another to form the integer L = 123456789101112131415 . . .. What is the smallest integer m > 2020 for which L is divisible by 9? Problem 5. Consider positive integers x and y. When y is divided by x the remainder is 29. When y is divided by x/2, the remainder is 13. Determine x. Problem 6. Find number of positive integers n such that n + 2n^2 + 3n^3 + . . . + 2019n^2019 is divisible by natural number (n-1). Problem 7. A farmer wants to buy 100 animals for $100.00 (must buy at least one of them)
Horses cost $5.00 each;
Cows cost $3.00 each;
Chickens cost $0.50 each. How many animals the farmer should buy ?
Problem 8. Prove that Problem 9. The product of the ages, in years, of three teenagers is 4590. None of the teens are the same age. What are the ages of the teenagers? Problem 10. The prime factorization of " k! " is Problem 11. If Problem 12. A sequence of digits is constructed by writing the digits of consecutive positive integers (123456789101112 ..... ). What is the digit in the 500th position? Problem 13. The pattern forming the irrational number 0.120210012000210000120000021... continues indefinitely. What is the 589-th digit in this pattern after the decimal point? Problem 14. If n is an integer between 1 and 96 (inclusive), find the probability that n(n+1)(n+2) is divisible by 8. Problem 15. Find six-digit multiples of 64 that consist only of the digits 2 and 4. Find the sum of digits of integer numbers Problem 1. Find the sum of the digits of all three-digit integer numbers Two-digit numbers with digit "9" Problem 1. A computer generates a two-digit integer random numbers. It can be any number from 00 to 99. Find the probability that it (a) - is 99; (b) - is not 99; (c) - has no 9s in it; (d) - has at least one 9 in it. Find a triangle with integer side lengths and integer area Problem 1. The lengths of the sides of a triangle are positive integers. One side has length 17 and the perimeter of the triangle is 54. If the area is also an integer, find the length of the longest side. Math circle level problem on the hundred-handed monster Briareus Problem 1. The hundred-handed monster Briareus was hungry and attacked a flock of 98 sheep. He didn't want to eat them all, so he developed a strange way to decide which ones to eat. With his hundred hands he lined the sheep up in front of him and attached tags labeling the sheep from 1 to 98, from left to right. First he crammed all the sheep into his mouth. Then he spat out all the even-numbered sheep. Then for every sheep with a number divisible by three, he stuffed it into his mouth if it was outside or spat it out if it was already inside. He did the same for every sheep with a number divisible by 4, 5, and so on up to 98. At the end, he devoured the sheep that were left in his mouth. How many sheep were left in the flock after Briareus left? Math Circle level problem on lockers and divisors of integer numbers Problem 1. There are 1000 students in a college. There are also 1000 lockers which are all opened. All the 1000 students are required to perform the following exercise: the first student should go round to change the state of all the lockers (that is if a locker is opened it should be closed, if a locker is closed, it should be opened). The second student should start with the second locker and change the state of all the lockers which are multiples of 2 (that is 2, 4, 6 etc. up to the 1000th locker). The third student should start with third fourth locker changing the state of all lockers which are multiples of 3 (that is all the lockers at 3, 6, 9, 12.... positions till all the 1000 lockers are attended to). The forth student should start with the fourth locker changing the state of all lockers which are multiples of 4 (that 4, 8, 12... till all the 1000 lockers are visited). The exercise should be done with all the students changing all the state of all lockers starting from their position and the multiples of lockers at their corresponding positions until the 1000th student change the state of the 1000th locker. At the end of the exercise, how many lockers remained closed? Nice entertainment problems related to divisibility properties Problem 1. Last year Jason's age was a prime number. This year it is a square number. How old is he this year? Problem 2. John thinks of two positive integers. He multiplies them together and then subtracts each of the integers from the product, with a result of 35. Find all possible pairs of numbers he could have chosen. Problem 3. In the sequence 456, 471, 483, 498, ... each term is equal to the previous term added to the sum of its digits. Which number of the following list does not belong to the sequence ? a) 1869; b) 1950; c) 3477; d) 4569; e) 5789. Problem 4. In the sequence 457, 473, 487, 506, ... each term is equal to the previous term added to the sum of its digits. Which number of the following list does not belong to the sequence ? a) 1864; b) 1949; c) 3466; d) 4569; e) 5767. Problem 5. Three girls, Ann, Betty and Cynthia, each have a younger brother, Dylan, Ernie and Frank, respectively. All six children do some fruit picking for the local farmer. The farmer agrees to pay each child as many dollars per basket as the number of baskets of fruit collected by that child. Each of the girls earned $45 more than her brother, and all six children collected a different number of baskets. How much did the farmer pay them all in total ? Problem 6. A bag contains red, white, green, and blue marbles. There are an equal number of red marbles and white marbles, and five times as many green marbles as blue marbles. There is a 35% chance of selecting a red marble first. What is the fewest possible number of green marbles in the bag? Problem 7. Assuming you have an unlimited supply of 3-cent and 7-cent stamps, what is the largest amount of postage you cannot make? Problem 8. Let A = For how many values of b is A an integer? Problem 9. Mike takes a tablet every 10 days. He has 25 tablets. If Mike took his first tablet on Monday, on what day will he take the last tablet? Solving problems on modular arithmetic Problem 1. Find the number of solutions to this system of three modular equations N = {2 mod 5}, N = {2 mod 6}, N = {2 mod 7}. Problem 2. Let x and y be integers. If x and y satisfy 41x + 5y = 31, then find the residue of x modulo 5. Problem 3. What is the inverse of 9 modulo 10? Problem 4. Find all integers n between 0 and 7 such that n is its own inverse modulo 8. Problem 5. Let m and n be non-negative integers. If m = 6n + 2, then what integer between 0 and m is the inverse of 2 modulo m? Problem 6. Find all integers Problem 7. I take a tablet every 10 days. If I take my first tablet on Monday and have 25 tablets, on what day will I take the last tablet ? Problem 8. Find all integer numbers x for which x^3 = (x - 1)^3 + (x - 2)^3 + (x - 3)^3 + (x - 4)^3 + (x - 5)^3 + (x - 6)^3 + (x - 7)^3. Problem 9. Find the number of subsets of S = {1, 3, 8, 17, 30, 36, 47, 58}, so that the sum of the elements in the subset is a multiple of 5. (Note that for the empty subset, we take the sum of the elements as 0.) Using the little Fermat's theorem to solve a problem on modular arithmetic Problem 1. For a certain positive integer n, the number What remainder does n leave when divided by 131? This lesson has been accessed 3395 times. |
