#P1933C. Turtle Fingers: Count the Values of k
Turtle Fingers: Count the Values of k
Description
You are given three positive integers $a$, $b$ and $l$ ($a,b,l>0$).
It can be shown that there always exists a way to choose non-negative (i.e. $\ge 0$) integers $k$, $x$, and $y$ such that $l = k \cdot a^x \cdot b^y$.
Your task is to find the number of distinct possible values of $k$ across all such ways.
The first line contains the integer $t$ ($1 \le t \le 10^4$) — the number of test cases.
The following $t$ lines contain three integers, $a$, $b$ and $l$ ($2 \le a, b \le 100$, $1 \le l \le 10^6$) — description of a test case.
Output $t$ lines, with the $i$-th ($1 \le i \le t$) line containing an integer, the answer to the $i$-th test case.
Input
The first line contains the integer $t$ ($1 \le t \le 10^4$) — the number of test cases.
The following $t$ lines contain three integers, $a$, $b$ and $l$ ($2 \le a, b \le 100$, $1 \le l \le 10^6$) — description of a test case.
Output
Output $t$ lines, with the $i$-th ($1 \le i \le t$) line containing an integer, the answer to the $i$-th test case.
11
2 5 20
2 5 21
4 6 48
2 3 72
3 5 75
2 2 1024
3 7 83349
100 100 1000000
7 3 2
2 6 6
17 3 632043
6
1
5
12
6
11
24
4
1
3
24
Note
In the first test case, $a=2, b=5, l=20$. The possible values of $k$ (and corresponding $x,y$) are as follows:
- Choose $k = 1, x = 2, y = 1$. Then $k \cdot a^x \cdot b^y = 1 \cdot 2^2 \cdot 5^1 = 20 = l$.
- Choose $k = 2, x = 1, y = 1$. Then $k \cdot a^x \cdot b^y = 2 \cdot 2^1 \cdot 5^1 = 20 = l$.
- Choose $k = 4, x = 0, y = 1$. Then $k \cdot a^x \cdot b^y = 4 \cdot 2^0 \cdot 5^1 = 20 = l$.
- Choose $k = 5, x = 2, y = 0$. Then $k \cdot a^x \cdot b^y = 5 \cdot 2^2 \cdot 5^0 = 20 = l$.
- Choose $k = 10, x = 1, y = 0$. Then $k \cdot a^x \cdot b^y = 10 \cdot 2^1 \cdot 5^0 = 20 = l$.
- Choose $k = 20, x = 0, y = 0$. Then $k \cdot a^x \cdot b^y = 20 \cdot 2^0 \cdot 5^0 = 20 = l$.
In the second test case, $a=2, b=5, l=21$. Note that $l = 21$ is not divisible by either $a = 2$ or $b = 5$. Therefore, we can only set $x = 0, y = 0$, which corresponds to $k = 21$.
In the third test case, $a=4, b=6, l=48$. The possible values of $k$ (and corresponding $x,y$) are as follows:
- Choose $k = 2, x = 1, y = 1$. Then $k \cdot a^x \cdot b^y = 2 \cdot 4^1 \cdot 6^1 = 48 = l$.
- Choose $k = 3, x = 2, y = 0$. Then $k \cdot a^x \cdot b^y = 3 \cdot 4^2 \cdot 6^0 = 48 = l$.
- Choose $k = 8, x = 0, y = 1$. Then $k \cdot a^x \cdot b^y = 8 \cdot 4^0 \cdot 6^1 = 48 = l$.
- Choose $k = 12, x = 1, y = 0$. Then $k \cdot a^x \cdot b^y = 12 \cdot 4^1 \cdot 6^0 = 48 = l$.
- Choose $k = 48, x = 0, y = 0$. Then $k \cdot a^x \cdot b^y = 48 \cdot 4^0 \cdot 6^0 = 48 = l$.