Lesson Problems on divisors of a given number
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<H2>Problems on divisors of a given number</H2> <H3>Problem 1</H3>Find the greatest 4 digit number that has exactly 3 factors. <B>Solution</B> <pre> If the number has exactly 3 factors, it means that the number is the square of a prime number: N = {{{p^2}}}. Then it has the factors 1 (one), p and {{{p^2}}}. Indeed, if the number is a prime number, it has only TWO factors: 1 (one) and itself. If the number is not prime and is not the square of a prime, then it has more than 3 factors. Therefore, to answer the problem's question, we must take the square of the largest two-digit prime number, which is {{{97^2}}} = 9409. <U>Answer</U>. The number under the question is 9409. </pre> <H3>Problem 2</H3>What is the smallest positive integer that has exactly 6 divisors ? <U>Solution</U>. <pre> 1. For integer number N = {{{p^alpha}}}, where p is a prime number and {{{alpha}}} is an integer exponent (index), the number of divisors is {{{alpha + 1}}}. You can easily check it: the divisors are 1, p, {{{p^2}}}, . . . , {{{p^alpha}}}. 2. For integer number N = {{{p^alpha*q^beta*ellipsis*r^theta}}}, where p, q, . . . , r are prime divisors and {{{alpha}}}, {{{beta}}}, . . . , {{{theta}}} are integer exponents (indexes) the number of divisors is {{{(alpha+1)*(beta+1)*ellipsis*(theta+1)}}}. 3. From these facts, you can easily obtain the answer. Notice that 6 = 2*3. It is easy to list those divisors: 1, 2, 4, 3, 6, 12. <U>Answer</U>. The smallest positive integer that has exactly 6 divisors is 12 = {{{2^2*3}}}. </pre> <H3>Problem 3</H3>Find all numbers between 200 and 500 which have exactly 9 factors. <B>Solution</B> <pre> A number, which has exactly 9 factors, is EITHER the 8th degree of a prime number OR is the square of an integer positive number " n ", which is the product of two prime numbers. Regarding the first case, the number can be only {{{2^8}}} = 256, which falls into the given interval. Regarding the second case, we look into the open interval from {{{sqrt(200)}}} ~ 14.1 to {{{sqrt(500)}}} ~ 22.4 and search there the integer numbers " n " that are the products of two different prime numbers. We find there ONLY THREE such numbers 15, 21 and 22 that produce the squares {{{15^2}}} = 225, {{{21^2}}} = 441, and {{{22^2}}} = 484. So, the <U>ANSWER</U> to the problem's question are the numbers 256, 225, 441 and 484. </pre> <H3>Problem 4</H3>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. <B>Solution</B> <pre> In order to facilitate your finding of the two distinct primes, notice that the given numbers represent an arithmetic progression with the common difference of 2997 - 1998 = 999 = 3*333 = 3*3*111 = 3*3*3*37. Next, since the number 1998 is divisible by 3 and by 37 (check it directly !), it implies that the given numbers ALL have 3 and 37 as their prime common divisors, and DO NOT HAVE ANY OTHER prime common divisors. Hence, the problem wants you find the sum of 3 and 37, which is 3+37 = 40. <U>ANSWER</U>. The sum of these two primes is 40. </pre> My other lessons in this site on miscellaneous problems on divisibility of integer numbers are - <A HREF=https://www.algebra.com/algebra/homework/divisibility/lessons/Light-flashes-on-a-Christmas-tree-and--a-Least-Common-Multiple.lesson>Light flashes on a Christmas tree and a Least Common Multiple</A> - <A HREF=http://www.algebra.com/algebra/homework/divisibility/lessons/The-number-that-leaves-a-remainder-1-when-divided-by-2-by-3-by-4-by-5-and-so-on-until-9.lesson>The number that leaves a remainder 1 when divided by 2, by 3, by 4, by 5 and so on until 9</A> - <A HREF=https://www.algebra.com/algebra/homework/divisibility/The-number-rem4-mod7-rem5-mod8-and-rem6-mod9.lesson>The number which gives remainder 4 when divided by 7, remainder 5 when divided by 8 and remainder 6 when divided by 9</A> - <A HREF=https://www.algebra.com/algebra/homework/divisibility/Introductory-problems-on-divisibility-numbers.lesson>Introductory problems on divisibility of integer numbers</A> - <A HREF=https://www.algebra.com/algebra/homework/divisibility/lessons/Finding-Greatest-Common-Divisor-of-integer-numbers.lesson>Finding Greatest Common Divisor of integer numbers</A> - <A HREF=http://www.algebra.com/algebra/homework/divisibility/lessons/Relativity-prime-numbers-help-to-solve-the-problem.lesson>Relatively prime numbers help to solve the problem</A> - <A HREF=https://www.algebra.com/algebra/homework/divisibility/lessons/Solving-equations-in-integer-numbers.lesson>Solving equations in integer numbers</A> - <A HREF=https://www.algebra.com/algebra/homework/divisibility/lessons/Quadratic-polynomial-with-odd-integer-coefs-can-not-have-a-rational-root.lesson>Quadratic polynomial with odd integer coefficients can not have a rational root</A> - <A HREF=https://www.algebra.com/algebra/homework/divisibility/Proving-the-equation-has-no-integer-solutions.lesson>Proving an equation has no integer solutions</A> - <A HREF=https://www.algebra.com/algebra/homework/divisibility/lessons/Composite-number-of-the-form-%284n%2B3%29-must--have-a-prime-divisor-of-the-form-%284n%2B3%29.lesson>Composite number of the form (4n+3) must have a prime divisor of the form (4n+3)</A> - <A HREF=https://www.algebra.com/algebra/homework/divisibility/lessons/How-many-three-digit-numbers-are-multiples-of-both-5-and-7.lesson>How many three-digit numbers are multiples of both 5 and 7?</A> - <A HREF=https://www.algebra.com/algebra/homework/divisibility/lessons/How-many-3-digit-numbers-are-not-dvsbl-by-2-not-dvsbl-by-3-notdvsbl-by-either-2-or-3.lesson>How many 3-digit numbers are not divisible by 2; not divisible by 3; not divisible by either 2 or 3</A> - <A HREF=https://www.algebra.com/algebra/homework/divisibility/lessons/How-many-integer-numbers-in-the-range-1-300-are-divisible-by.lesson>How many integer numbers in the range 1-300 are divisible by at least one of the integers 4, 6 and 15 ?</A> - <A HREF=https://www.algebra.com/algebra/homework/divisibility/lessons/Find-the-remainder-of-division.lesson>Find the remainder of division </A> - <A HREF=https://www.algebra.com/algebra/homework/divisibility/lessons/Why-3%5En%2B7%5En-2-is-divisible-by-8-for-all-positive-integer-n.lesson>Why 3^n + 7^n - 2 is divisible by 8 for all positive integer n ?</A> - <A HREF=https://www.algebra.com/algebra/homework/divisibility/What-is--the-last-digit-of-the-number-a%5En.lesson>What is the last digit of the number a^n ?</A> - <A HREF=https://www.algebra.com/algebra/homework/divisibility/lessons/Find-the-last-three-digits-of-these-numbers.lesson>Find the last three digits of these numbers</A> - <A HREF=https://www.algebra.com/algebra/homework/divisibility/lessons/Find-the-two-last-digits-of-the-number-3%5E123%2B7%5E123%2B9%5E123.lesson>Find the last two digits of the number 3^123 + 7^123 + 9^123</A> - <A HREF=https://www.algebra.com/algebra/homework/divisibility/Find-the-last-two-digits-of-%281%21-%2B-2%21-%2B-3%21-%2B-%2B-2024%21%29%5E2024.lesson>Find the last two digits of (1! + 2! + 3! + ... + 2024!)^2024</A> - <A HREF=https://www.algebra.com/algebra/homework/divisibility/lessons/Find-n-th-term-of-a-sequence.lesson>Find n-th term of a sequence</A> - <A HREF=https://www.algebra.com/algebra/homework/divisibility/lessons/Solving-one-Diofantine-equation.lesson>Solving Diophantine equations</A> - <A HREF=https://www.algebra.com/algebra/homework/divisibility/lessons/Determine-the-number-of-integer-solutions-to-equation-n%5E2%2B18n%2B3=m%5E2.lesson>How many integers of the form n^2 + 18n + 13 are perfect squares</A> - <A HREF=https://www.algebra.com/algebra/homework/divisibility/lessons/Miscellaneous-problems-on-divisibility-numbers.lesson>Miscellaneous problems on divisibility numbers</A> - <A HREF=https://www.algebra.com/algebra/homework/divisibility/lessons/Find-the-sum-of-digits-of-integer-numbers.lesson>Find the sum of digits of integer numbers</A> - <A HREF=https://www.algebra.com/algebra/homework/divisibility/lessons/Two-digit-numbers-with-digit-9.lesson>Two-digit numbers with digit "9"</A> - <A HREF=https://www.algebra.com/algebra/homework/divisibility/lessons/Find-a-triangle-with-integer-side-lengths-and-integer-area.lesson>Find a triangle with integer side lengths and integer area</A> - <A HREF=https://www.algebra.com/algebra/homework/divisibility/lessons/Math-circle-level-problem-on-the-hundred-handed-monster-Briareus.lesson>Math circle level problem on the hundred-handed monster Briareus</A> - <A HREF=https://www.algebra.com/algebra/homework/divisibility/lessons/Math-Citcle-level-problem-on-lockers-and-divisors.lesson>Math Circle level problem on lockers and divisors of integer numbers</A> - <A HREF=https://www.algebra.com/algebra/homework/divisibility/lessons/Nice-entertaiment-problem-What-numbers-John-is-thinking-about.lesson>Nice entertainment problems related to divisibility property</A> - <A HREF=https://www.algebra.com/algebra/homework/divisibility/lessons/Solving-problems-on-modular-arithmetic.lesson>Solving problems on modular arithmetic</A> - <A HREF=https://www.algebra.com/algebra/homework/divisibility/lessons/Using-the-little-Fermat-theorem-to-solve-a-problem-on-modular-arithmetic.lesson#google_vignette>Using the little Fermat's theorem to solve a problem on modular arithmetic</A> - <A HREF=https://www.algebra.com/algebra/homework/divisibility/lessons/OVERVIEW-of-miscellanious-solved-problems-on-divisibility-of-numbers.lesson>OVERVIEW of miscellaneous solved problems on divisibility of integer numbers</A>
