1. Find the second smallest positive integer that gives a remainder of 2 when divided by 3 and gives a remainder of 3 when divided by 7.
2. When a positive integer is expressed in base 7, it is AB_7, and when it is expressed in base 5, it is BA_5. What is the positive integer in decimal?
3. What is the remainder when 1^3+2^3+3^3+...+100^3 is divided by 6?
4. There are finitely many primes p for which the congruence \(8x\equiv 1\pmod{p}\) has no solutions x. Determine the sum of all such p.
+129840 1. Find the second smallest positive integer that gives a remainder of 2 when divided by 3 and gives a remainder of 3 when divided by 7.
We have that
N = 3a + 2
N = 7b + 3
Subtracting these equations we have that
3a - 7b - 1 = 0
3a - 7b = 1
And the second smallest positive integer results when a = 12 and b = 5
So.....the second smallest integer is 38.....just as Max said !!!!
+9665 1) We first have a list of what gives a remainder 2 when divided by 3 and we will have a list of what gives a remainder of 3 when divided by 7.
Gives a remainder 2 when divided by 3:
2,5,8,11,14,17,20,.......
Gives a remainder 3 when divided by 7:
3,10,17,......
Then we add 17 to the LCM of 3 and 7:
17 + 21 = 38
Answer : 38
+129840 1. Find the second smallest positive integer that gives a remainder of 2 when divided by 3 and gives a remainder of 3 when divided by 7.
We have that
N = 3a + 2
N = 7b + 3
Subtracting these equations we have that
3a - 7b - 1 = 0
3a - 7b = 1
And the second smallest positive integer results when a = 12 and b = 5
So.....the second smallest integer is 38.....just as Max said !!!!
