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Karatsuba's basic step works for any base B and any m, but the recursive algorithm is most efficient when m is equal to n/2, rounded up. In particular, if n is 2 k, for some integer k, and the recursion stops only when n is 1, then the number of single-digit multiplications is 3 k, which is n c where c = log 2 3.
Karatsuba multiplication is an O(n log 2 3) ≈ O(n 1.585) divide and conquer algorithm, that uses recursion to merge together sub calculations. By rewriting the formula, one makes it possible to do sub calculations / recursion. By doing recursion, one can solve this in a fast manner.
If they are 00, do nothing. Use P directly in the next step. If they are 11, do nothing. Use P directly in the next step. Arithmetically shift the value obtained in the 2nd step by a single place to the right. Let P now equal this new value. Repeat steps 2 and 3 until they have been done y times. Drop the least significant (rightmost) bit from P.
Long division is the standard algorithm used for pen-and-paper division of multi-digit numbers expressed in decimal notation. It shifts gradually from the left to the right end of the dividend, subtracting the largest possible multiple of the divisor (at the digit level) at each stage; the multiples then become the digits of the quotient, and the final difference is then the remainder.
The above equations confirm that there are no other Kaprekar's constants than 495 and 6174. There are no Kaprekar numbers for 1, 2, 5, or 7 digits, since they do not satisfy any of equations (1)~(5). For six-digit numbers, there are two solutions that satisfy equations (1) and (2). [9]
The input section is shifted one digit to the left with the end crank. The next digit of the multiplier is set into the multiplier dial, and the crank is turned again, multiplying the operand by that digit and adding the result to the accumulator. The above two steps are repeated for each multiplier digit.
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In binary (base-2) math, multiplication by a power of 2 is merely a register shift operation. Thus, multiplying by 2 is calculated in base-2 by an arithmetic shift. The factor (2 −1) is a right arithmetic shift, a (0) results in no operation (since 2 0 = 1 is the multiplicative identity element), and a (2 1) results in a left arithmetic shift ...
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related to: shift solve calculator with steps 2 and 6 digit