BEHOLD! decimal numeration imposes patterns of tens; patterns of tens of tens (hundreds); patterns of tens of tens of tens (thousands); etc.BEBEHOLD! 10 = 2 x 5, which invokes the 2-ness and 5-ness subpatterns of ten pattern imposed.
Elsewhere, you see the POWERS OF 2 PATTERNS: the 2-ness subpatterns of ten pattern imposed. Now, I'll show you the POWERS OF 5 PATTERNS: the 5-ness subpatterns of ten pattern imposed. To see this, you color (say) RED the 5-multiples -- 0, 5, 10, 15, ..., 95.
This yields an alternating pattern (similar to the alternating pattern for 2-ness), which I "stain"" as RED (5-multiple)-columns and NONRED (non-5-multiple)-columns.
Seeing, the child can be be guided to a Conclusion:The 5-ness (non-5-ness) pattern repeats every 10. Then the child can be guided to see that, equivalently, 5-ness (non-5-ness) is conserved by the transformation of adding 10.
This also means (as you know, and the child learns in Elementary Arithmetic Classes): "A number is a 5-multiple if, and only if, its last digit is a multiple of 5."
Next, you color (say) blue the 25-multiples (that is, 5-ness of 5-ness).From the coloring, the child can be guided to see that the 25-ness forms another pattern which repeats every 100 (10-ness of 10-ness). And guided to see the 25-ness (or non-25-ness) pattern is conserved by the transformation of adding 100 to a number. This also means (something rarely, if ever taught!) that a number is a 25-multiple if, and only if, the last 2 digits form a 25-multiple.
Thus, 35775 is a 25-multiple because its last two digits, 75, form a 25-mlutiple.
That's all you need to check to detect 25-multipleness -- just check the last 2 digits.
It is tedious to show the patterns for 125 = 5 x 5 x 5. It jumps from 0 to 125 to 250 to 375 to 500 to 625 to 750 to 875 to 1000, then repeats the pattern over with 10125, 1250, 1375, 1500, 1625, 1750, 1875, 2000, and repeats this again, etc.Equivalently, 125-ness (non-125-ness) is conserved by the transformation of adding 1000.
And the child can be guided to realize (something rarely, if ever taught!) that A number is an 125-multiple if, and only if, its last 3 digits form an 125-mlutiple.
Thus, 359750 is an 125-multiple because its last 3 digits, 750, form an 125-multiple.
Also, it would be tedious to carry out the next 625-step, since it skips 0, 625, 1250, 1875, 2500, 3125, 3750. 4375, 5000, 5625, 6250, 6875, 7500, 8125, 8750, 9375, 10000 -- then repeats this pattern.Thus, the child could be guided to see that 625-ness (5 x 5 x 5 x 5) repeats every 10,000 (10 x 10 x 10 x 10). Equivalently, 625-ness (non-125-ness) pattern is conserved by adding 10,000. And this means (ever taught?) that a number is a 625-multiple if, and only if, its last 4 digits forms a 625-multiple.
Thus, 3576250 is a 625-multiple because its last 4 digits (6250) form a 625-multiple.
In general, the n5-ness pattern (multiplying n 5's) repeats every 10n (multiplying n 10's). The child can be told that this INVOKES A 5-POWER CONSERVATION LAW: a number is a multiple of an n-product of 5's if, and only if, its last n digits forms a multiple of an n-product of 5's.