Lesson Find the last two digits of the number 3^123 + 7^123 + 9^123
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<H2>Find the last two digits of the number 3^123 + 7^123 + 9^123</H2> <H3>Problem 1</H3>Find the last two digits in the decimal numeral for {{{(3^167)^95}}}. <B>Solution</B> <pre> (1) First, let's find two last digits of the number {{{3^167}}}. It is clear, that looking for the last two digits of the number {{{3^167}}}, we should track last two digits of the number of {{{3^k}}} for k = 1, 2, 3, 4, . . . This sequence of the last two digits of numbers {3^k mod 100} is periodical. The period starts from the first term and the period length is 20, as it can be seen from this table below k 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 3^k mod 100 3 9 27 81 43 29 87 61 83 49 47 41 23 69 7 21 63 89 67 1 3 Since 167 = 8*20 + 7, the last two digits of the number {{{3^167}}} is the 7th term of this cyclic sequence, i.e. the number 87. (2) So, now next goal is to determine the last two digits of the number {{{87^95}}}. Again, it is clear, that looking for the last two digits of the number {{{87^95}}}, we should track last two digits of the numbers {{87^k}}} for k = 1, 2, 3, 4, . . . This sequence of the last two digits of numbers {87^k mod 100} is periodical. The period starts from the first term and the period length is 20, as it can be seen from this table below k 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 87^k mod 100 87 69 3 61 7 9 83 21 27 49 63 81 47 89 43 21 67 29 23 1 87 Since 95 = 4*20 + 15, the last two digits of the number {{{87^95}}} is the 15th term of this cyclic sequence, i.e. the number 43. <U>ANSWER</U>. The last two digits in the decimal numeral for {{{(3^167)^95}}} are 43. </pre> <H3>Problem 2</H3>Find the last two digits of the number 3^123 + 7^123 + 9^123. <B>Solution</B> <pre> To find the last two digits of the number 3^123 + 7^123 + 9^123, it is enough to know the last two digits of each number 3^123, 7*123 and 9^123, separately. The last two digits of the numbers 3^1, 3^2, 3^3, . . . , 3^k, . . . repeat periodically with the period of 20. So (and in particular), two last digits of the numbers 3^21, 3^41, . . . , 3^121 are the same: they are 03. The two last digits of the number 3^123, therefore, is not difficult to calculate : they are 27. The last two digits of the numbers 7^1, 7^2, 7^3, . . . , 7^k, . . . repeat periodically with the period of 4. So (and in particular), two last digits of the numbers 7^5, 7^9, . . . , 7^121 are the same: they are 07. The two last digits of the number 7^123, therefore, is not difficult to calculate : they are 43. The last two digits of the numbers 9^1, 9^2, 9^3, . . . , 9^k, . . . repeat periodically with the period of 10. So (and in particular), two last digits of the numbers 9^11, 9^21, . . . , 9^121 are the same: they are 09. The two last digits of the number 9^123, therefore, is not difficult to calculate : they are 29. Therefore, the last two digits of the number 3^123 + 7^123 + 9^123 you can easily find by taking the sum 27 + 43 + 29 = 99. <U>ANSWER</U>. The last two digits of the number 3^123 + 7^123 + 9^123 are 99. </pre> Solved. ------------------ <U>A post-solution note</U> <pre> The fact that the last two digits of the sequence 3, 3^2, 3^3, . . . , 3^k . . . form a periodic sequence seems to be a miracle. But this fact is INEVITABLE consequence of simple properties. If to consider the last two digits, there are only finite number of their combinations: 00, 01, 02, . . . 10, 11, 12, . . . , 98, 99 --- in all, there are only 100 such 2-digit combinations. From the other side, the sequence 3, 3^2, 3^3, . . . , 3^k . . . is INFINITE. Therefore, by projecting it into the last two digits sequence, we INEVITABLY will have a repetition; and as soon as such a repetition will happen for the first time, the periodic behavior and the period itself are just provided. So, there is no any miracle in it - it is an inevitable fact. An only heuristic is from which term does the period start and what is the length of the period? These questions are easily to find out making an Excel spreadsheet - then it can be found out in seconds. In the given problem, periods start from the first term and the lengths of the periods were stated above. The facts I explained in this lesson and in the post-solution note, play a KEY ROLE in similar proofs, making a key for solving such problems. Those who study Math from Math schools and/or from Math circles (or from associated Math books), usually know this remarkable property. Now, after reading this lesson, you know this property, too (!) So from now, you do belong to a category of those lucky persons who know it (!) My congratulations (!) (!) </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/Problems-on-divisors-of-a-given-number.lesson>Problems on divisors of a given number</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/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>
