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An arrangement of D flip-flops is a classic method for integer-n division. Such division is frequency and phase coherent to the source over environmental variations, including temperature. The easiest configuration is a series where each D flip-flop is a divide-by-2. For a series of three of these, such a system would be a divide-by-8.
An asynchronous (ripple) counter is a "chain" of toggle (T) flip-flops in which the least-significant flip-flop (bit 0) is clocked by an external signal (the counter input clock), and all other flip-flops are clocked by the output of the nearest, less significant flip-flop (e.g., bit 0 clocks the bit 1 flip-flop, bit 1 clocks the bit 2 flip ...
In binary arithmetic, division by two can be performed by a bit shift operation that shifts the number one place to the right. This is a form of strength reduction optimization. For example, 1101001 in binary (the decimal number 105), shifted one place to the right, is 110100 (the decimal number 52): the lowest order bit, a 1, is removed.
Flip-Flops that read in a new value on the rising and the falling edge of the clock are called dual-edge-triggered flip-flops. Such a flip-flop may be built using two single-edge-triggered D-type flip-flops and a multiplexer, or by using two single-edge triggered D-type flip-flops and three XOR gates.
At each advance, the bit on the far left (i.e. "data in") is shifted into the first flip-flop's output. The bit on the far right (i.e. "data out") is shifted out and lost. The data is stored after each flip-flop on the "Q" output, so there are four storage "slots" available in this arrangement, hence it is a 4-bit register.
The output of a flip-flop is constant until a pulse is applied to its "clock" input, upon which the input of the flip-flop is latched into its output. In a synchronous logic circuit, an electronic oscillator called the clock generates a string (sequence) of pulses, the "clock signal".
The JK flip-flop augments the behavior of the SR flip-flop (J=Set, K=Reset) by interpreting the J = K = 1 condition as a "flip" or toggle command. This is an important distinction as it would confuse me to see S = R = 1 in the introduction sentence and then to read the Characteristic table, which has J = K = 1 for the flip.
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.