E. Novice's Mistake

One of the first programming problems by K1o0n looked like this: "Noobish_Monk has 𝑛 (1≤𝑛≤100) friends. Each of them gave him 𝑎 (1≤𝑎≤10000) apples for his birthday. Delighted with such a gift, Noobish_Monk returned 𝑏 (1≤𝑏≤min(10000,𝑎⋅𝑛)) apples to his friends. How many apples are left with Noobish_Monk?"

K1o0n wrote a solution, but accidentally considered the value of 𝑛 as a string, so the value of 𝑛⋅𝑎−𝑏 was calculated differently. Specifically:

• when multiplying the string 𝑛 by the integer 𝑎, he will get the string 𝑠=𝑛+𝑛+⋯+𝑛+𝑛⏟𝑎 times
• when subtracting the integer 𝑏 from the string 𝑠, the last 𝑏 characters will be removed from it. If 𝑏 is greater than or equal to the length of the string 𝑠, it will become empty.

Learning about this, ErnKor became interested in how many pairs (𝑎,𝑏) exist for a given 𝑛, satisfying the constraints of the problem, on which K1o0n's solution gives the correct answer.

"The solution gives the correct answer" means that it outputs a non-empty string, and this string, when converted to an integer, equals the correct answer, i.e., the value of 𝑛⋅𝑎−𝑏.