Hi Rick,
There isn't a Ila, Idaho today. Therefore I don't know what county it was in.

Her first name of Aito is extremely unusual and I didn't find anything in the Census records for her.

However I did find an Arta Scroggin in 1900 living in Genessee, Latah Co., ID.
Archer R. Scroggin, b. IL June 1859, marr 16 yrs, bank cashier.
Mary E., wife, b. IL, Jan 1856, had given birth to eight children but only four alive in 1900.
Arta L., dau, b. MO Feb 1888.
Carrie M., b. OR Oct 1890.
Fanny M., b. ID July 1892.
Grace B., b. ID Jan 1898.
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