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I'll start off by stating that I am a novice.

I'm trying to figure out how to use a decade counter, so I've constructed this circuit: enter image description here

It worked fine, but now I'm wondering how it was decided to use a 47kΩ resistor on pin 15 and a 10kΩ resistor on pin 13?

How would I calculate these required resistor values in a different setting?

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Pin 15 is the Reset input. It is set high to reset the count to '0'. Pin 13 is the Clock Enable input. It must be set low to accept pulses on the Clock input.

CMOS logic inputs will 'float' to an undefined voltage if not pulled high or low. Resistors are commonly used when the signal can only pull one way (eg. a pushbutton or diode). In your circuit both of these pins could be connected directly to ground, so the resistors are redundant. However they might be necessary if you wanted to reset the counter or disable counting, and then the values would depend on how that was done.

I am guessing that this circuit was copied from another one which reset the counter on power up and/or fewer than 10 counts, and had a clock enable input. Whoever 'designed' this version probably just slavishly copied the resistor values because they didn't know their purpose.

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I don’t see any issue. The two pull-down resistors are on unused pins. Their values don’t matter at all as it’s a CMOS device which has practically no input current. 10K is fine. 47K is fine. So is tying them them directly to ground.

What you cannot do is leave them floating. All CMOS inputs must be driven by something.

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