Lesson What is the last digit of the number a^n ?
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<H2>What is the last digit of the number a^n ?</H2> <H3>Problem 1</H3>What is the last digit of the number {{{7^72}}} ? <B>Solution</B> <pre> To solve this problem, you do not need calculate the numbers {{{7^n}}}, one after another. It is enough to trace the LAST DIGITS of these numbers, only. Last digits of {{{7^n}}} form the sequence n 1 2 3 4 5 . . . 72 last digits of {{{7^n}}} 7 9 3 1 7 . . . (1) From this Table, the first four terms of this sequence are different numbers 7, 9, 3, 1, but then the digit 7 arises again as the 5-th term. From this point, it should be clear to you, that this sequence is periodic with the period length of 4. The number of periods of the length 4 in the sequence (1) is exactly {{{72/4}}} = 18, so the last digit in this sequence is the last digit of the period {7, 9, 3, 1}. Thus the last digit of the number {{{7^72}}} is 1. <U>ANSWER</U>. {{{7^72}}} has the last digit 1. </pre> <H3>Problem 2</H3>What is the units digit of the number {{{19^93}}} ? <B>Solution</B> <pre> To solve this problem, you do not need calculate the numbers {{{19^n}}}, one after another. It is enough to trace the LAST DIGITS of these numbers, only. Last digits of {{{19^n}}} form the sequence n 1 2 3 4 5 . . . last digits of {{{19^n}}} 9 1 9 1 9 . . . (2) From this Table, the first two terms of this sequence are different numbers 9, 1, but then the digit 9 arises again as the 3-rd term. From this point, it should be clear to you, that this sequence is periodic with the period length of 2. The number of full periods of the length 2 in the sequence (2) is the integer part of the number {{{93/2}}}, i.e. 46. so the last digit in the sequence (2) is the first digit of the next period {9, 1}, i.e. 9. Thus {{{19^93}}} has the last digit 9. <U>ANSWER</U>. {{{19^93}}} has the last digit 9. </pre> <H3>Problem 3</H3>Determine the ONES digit of the number {{{23^85}}}. <B>Solution</B> <pre> The ones digits of consecutive powers of the number 23 form this sequence n 1 2 3 4 5 6 7 8 9 10 The last digit of {{{23^n}}} 3 9 7 1 3 9 7 1 3 9 The sequence of the last digits is periodical. The period starts from n = 1 and has the length of 4. 85 = 4*21 + 1. Therefore, the last digit of the number {{{23^85}}} is equal to 1 : the first number of the period. <U>ANSWER</U> </pre> <H3>Problem 4</H3>Find the last digit of the sum {{{3^2018}}} + {{{4^2018}}}. <B>Solution</B> <pre> n : 1 2 3 4 5 6 7 8 9 10 Last digit of {{{3^n}}} : 3 9 7 1 3 9 7 1 3 9 Last digit of {{{4^n}}} : 4 6 4 6 4 6 4 6 4 6 The last digits of the number {{{3^n}}} form a periodical sequence. The period starts from n= 1 and has the length of 4. 2018 = 504*4 + 2. So the number {{{3^2018}}} has the last digit 9: the second digit in the cycle. The last digits of the number {{{4^n}}} form a periodical sequence. The period starts from n= 1 and has the length of 2. 2018 = 1008*2 + 2. So the number {{{4^2018}}} has the last digit 6: the second digit in the cycle. Therefore, the sum {{{3^2018}}} + {{{4^2018}}} has the last digit 9 + 6 = 5 (mod 10). <U>ANSWER</U> </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/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>
