franloranca wrote:
Bunuel wrote:
honchos wrote:
Which of the following numbers is prime?
A. 2^16+1
B. 2^31+3^31
C. 4^66+7^66
D. 5^82−2^82
E. 5^2881+7^2881
Let's check which of the options is NOT a prime:
A. 2^16 + 1 --> the units digit of 2 in positive integer power repeats in blocks of four {2, 4, 8, 6}. Hence, the units digit of 2^16 is 6 and the units digit of 2^16 + 1 is 7 --> 2^16 + 1 CAN be a prime.
B. 2^31 + 3^31 --> the units digit of 2^31 is 8 and the units digit of 3^31 is 7 (the units digit of 3 in positive integer power repeats in blocks of four {3, 9, 7, 1}). Hence, the units digit of 2^31 + 3^31 is 5 (8+7). Thus 2^31 + 3^31 is divisible by 5. Not a prime.
C. 4^66 + 7^66 --> the units digit of 4^66 is 6 (the units digit of 4 in positive integer power repeats in blocks of two {4, 6}) and the units digit of 7^66 is 9 (the units digit of 7 in positive integer power repeats in blocks of four {7, 9, 3, 1}). Hence, the units digit of 4^66 + 7^66 is 5 (6+9). Thus 4^66 + 7^66 is divisible by 5. Not a prime.
D. 5^82 - 2^82 --> we can factor this as (5^41 - 2^41)(5^41 + 2^41). Not a prime.
E. 5^2881+ 7^2881 --> 5^2881 + 7^2881 = odd + odd = even. Not a prime.
Only option A can be prime.
Answer: A.
Hope it's clear.
Bunuel,
..... C. 4^66 + 7^66 --> the units digit of 4^66 is 6 (the units digit of 4 in positive integer power repeats in blocks of two {4, 6})..... 66/2 (number of repetitions of two) is 33, and therefore it should be the first number of the block of 2, meaning the units digits is 4. I think I am missing something
Consider the following example: what is the units digit of 127^124.
First of all, the units digit of 127^124 is the same as that of 7^124 (get rid of all the digits except the units digit).
Next, recall that the units digit of 7 in positive integer power repeats in blocks of four {7, 9, 3, 1}.
Finally, to get the units digit of 7^124, you need to divide the exponent (124) by 4 (cyclicity) and look at the remainder you get:
Remainder = 1 --> the units digit = 1st number from the pattern, so 7.
Remainder = 2 --> the units digit = 2nd number from the pattern, so 9.
Remainder = 3 --> the units digit = 3rd number from the pattern, so 3.
Remainder = 0 --> the units digit = 4th number from the pattern, so 1.
Now, since 124/4 yields the remainder of 0 (124 is divisible by 4), then the units digit of 7^124 is 1.
We can apply the same logic to 4^66: the units digit of 4 in positive integer power repeats in blocks of two {4, 6} --> 66/2 yields the remainder of 0, thus the units do digit of 4^66 is 2nd number from the pattern, so 6. Or another way: 4^odd has the units digit of 4 and 4^even has the units digit of 6.
For more check Number Theory chapter of our Math Book:
math-number-theory-88376.htmlHope it's clear.
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