Shave off Last Two

One can test if an integer nn is divisible by prime 101101 by subtracting the last two digits (as a number) from nn with those digits shaved off, and see if the result is divisible by 101101. For example, 162794931162794931 is divisible by 101101 because 162794931=16279181627949 - 31 = 1627918 and 1011627918101 \, | \, 1627918.

What are the smallest factors you should multiply the last two digits of nn (as a number) with, before subtracting from the shaved-off nn in the divisibility tests for p=43p = 43 and p=67?p = 67?

Give your answer as the product of the factors.

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