A series if digits
0=0 in binary 1=1 2=10 3=11 . . . Got it?
No. 740 → 7 + 4 + 0 = 11 11 → 1 + 1 = 2 which is not one of 3, 6, 9 so 740 is not a multiple of 3.
14: Quarters Dimes Nickels 0 0 13 0 1 11 0 2 9 0 3 7 0 4 5 0 5 3 0 6 1 1 0 8 1 1 6 1 2 4 1 3 2 1 4 0 2 0 3 2 1 1
A huge number. 0 + 1 + 2 = 3 0 + 2 + 1 = 3 1 + 0 + 2 = 3 1 + 2 + 0 = 3 2 + 0 + 1 = 3 2 + 1 + 0 = 3 -0 + 1 + 2 = 3 -0 + 2 + 1 = 3 1 - 0 + 2 = 31 + 2 - 0 = 32 - 0 + 1 = 32 + 1 - 0 = 3 0 - 1 + 3 = 2 0 + 3 - 1 = 2 -1 + 0 + 3 = 2 -1 + 3 + 0 = 2 3 + 0 - 1 = 2 3 - 1 + 0 = 2 -0 - 1 + 3 = 2-0 + 3 - 1 = 2-1 - 0 + 3 = 2-1 + 3 - 0 = 23 - 0 - 1 = 23 - 1 - 0 = 2 0 - 2 + 3 = 1 0 + 3 - 2 = 1 -2 + 0 + 3 = 1 -2 + 3 + 0 = 1 3 + 0 - 2 = 1 3 - 2 + 0 = 1 -0 - 2 + 3 = 1-0 + 3 - 2 = 1-2 - 0 + 3 = 1-2 + 3 - 0 = 13 - 0 - 2 = 13 - 2 - 0 = 1 1 + 2 - 3 = 0 1 - 3 + 2 = 0 2 + 1 - 3 = 0 2 - 3 + 1 = 0 -3 + 1 + 2 = 0 -3 + 2 + 1 = 0 For each of these equations there is a counterpart in which all signs have been switched. For example 0 + 1 + 2 = 3 gives -0 - 1 - 2 = -3and so on. Now, all of the above equations has three numbers on the left and one on the right. Each can be converted to others with two numbers on each side. For example:the equation 0 + 1 + 2 = 3 gives rise to0 + 1 = 3 - 20 + 1 = -2 + 30 + 2 = 3 - 10 + 2 = -1 + 31 + 2 = 3 - 01 + 2 = -0 + 3-0 + 1 = 3 - 2-0 + 1 = -2 + 3-0 + 2 = 3 - 1-0 + 2 = -1 + 31 + 2 = 3 + 01 + 2 = +0 + 3 As you can see, the number of equations is huge!
They are 12, 11, 10, ..., 2, 1, 0, -1, -2, -3, ...
1, 3, 11, 33, 61, 183, 671, 2013 3, 11 and 61 are prime.
11. 0 = 0 1 = 1 2 = 10 3 = 11
x = -1/3 or 11/2 3 = 6x2 - 7x ⇒ 6x2 - 7x - 3 = 0 ⇒ (3x + 1)(2x - 3) = 0 ⇒ (3x + 1) = 0 → x = -1/3 or (2x - 3) = 0 → x = 11/2
1, 3, 11, 33, 61, 183, 671, 2013
1, 3, 11, 33, 61, 183, 671, 2013.
1, 2, 11, 22 1, 3, 19, 57 1, 61 1, 97
1, 3, 11, 33, 61, 183, 671, 2013
0=0 in binary 1=1 2=10 3=11 . . . Got it?
They are the 62 terms of the sequence: t(n) = 61!/[n1*(61-n)!] for n = 0, 1, 2, ... , 61. n! denotes a*2*3* ... *n
11
The prime numbers from 1 to 61 are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, and 61.
2x - 3y = 4 x + 4y = 3 Write the augmented matrix: 2 -3 4 1 4 3 Multiply the first row by 1/2: 1 -3/2 2 1 4 3 Subtract the second row from the first row: 1 - 3/2 2 0 -11/2 -1 Multiply the second row by -2/11: 1 -3/2 2 0 1 2/11 Multiply the second row by 3/2 and add it to the first row: 1 0 25/11 0 1 2/11 Thus x = 25/11 and y = 2/11 Check: