What will be the unit digit in 3^63?

How do you determine composite number of 63? The easiest way to tell if 63 ia a composite number is to determine if 3 will divide evenly into the digits when they are added together. 63 6 + 3 = 9. 9 is divisible by 3. 63 is a composite number. If you want to know what composite number will divide evenly into 63, you can list the factor pairs: 1 x 63 3 x 21 7 x 9 1, 3, and 7 are prime numbers. 9, 21, and 63 are the factors of 63 that are composite numbers.

Note that the last digit of a positive integer is the remainder when you divide it by $10$. Perhaps you have noted that $a^b$ and $a^$ ends in the same digit as $7^3$, that is, $3$. Consider $7^k$ modulo $10$. You will notice that it enters a repeating pattern: $7,9,3,1,7,9,3,1,7,9,3,1,\dots$ Where $7^k\mod 10 \equiv \begin$ Now, consider the exponent in this case. $8^7$. Since the pattern of $7^k$ depends on $k\mod 4$, what is $8^7\mod 4$? $\begingroup$ @Heidi for $9^k$ you will notice that it follows the pattern $9,1,9,1,9,1,\dots$, so you see that $9^\equiv 9\mod 10$. Whatever the base is, you try to find out how frequently the pattern repeats (it will always eventually repeat), and find out how the exponent fits into that pattern. $\endgroup$ I was just wondering why the ajotatxe's answer below doesn't apply to this case 7^(6^21). In the general case, you must note also that the remainder when divided by 4 of the sucesive powers of a number present also this "periodic behaviour". Take for example 7119 We need the remainder of 119:4, but the remainder of 11:4, 112:4, 113:4, etc is 1 for even powers and 3 for odd powers; therefore, 7119 ends in the same digit as 73, that is, 3. Following the same logic to 7^(6^21) we have 6^21 has unit digit equal to 6, because every number ending in 6 powered to any number is 6. So the second step is to solve 7^6 which is 9 right? 9 is the wrong answer, the true result is 1! you can check it here Thanks

Problem 1 : What is the unit digit in the product of (684 x 759 x 413 x 676) ? Solution : Product of unit digits of the given whole numbers is = (4 x 9 x 3 x 6) = 36 x 18 Unit digit of the product = 8. So, the unit digit of the product of given whole numbers is 8. Problem 2 : What is the unit digit in the product (3547) 153 x (251) 72 ? Solution : Unit digit of 3547 is 7 Evaluating 7 153 : 7 1 = 7 (Unit digit is 7) 7 2 = 49 (Unit digit is 9) 7 3 = 343 (Unit digit is 3) 7 4 = 2401 (Unit digit is 1) 7 5=2401 x 7 (Unit digit is 7) Every cycle consists of interval 4. By dividing 153 by 4, we get 1 as remainder. So, the unit digit of 7 153 is 7. Unit digit of 251 is 1 Evaluating 1 72 : Unit digit of 1 72 is 1. So, the unit digit of the given product is 7. Problem 3 : What is the unit digit in 264 102 + 264 103 ? Solution : = 264 102+ 264 103 = 264 102 (1 + 264) = 264 102 (265) Calculating the cyclicity of 4 : 4 1 = 4 4 2 = 16 4 3 = 64 4 4 = 256 Every cycle consists of interval 2.By dividing 102 by 2, we will get 0 as remainder. So, the unit digit of 4 102 is 6. 6(5) = 30 (unit digit is 0) So, the required unit digit is 0. Problem 4 : What is the unit digit of 7 95 - 3 58 ? Solution : Cyclicity of 7 : 7 1= 7 (Unit digit is 7) 7 2 = 49(Unit digit is 9) 7 3 = 343(Unit digit is 3) 7 4= 2401(Unit digit is 1) 7 5= 2401 x 7 (Unit digit is 7) Every cycle consists of interval 4. By dividing 95 by 4, we get 3 as remainder. According to cyclicity of 7, 3 will be the unit digit. Cyclicity ...