Hello, I am having difficulty understanding this topic. I understand that the important thing to look for is the “limiting reactant” in the two factoring equations, which is 7, but won’t the denominator of QB always be inherently bigger than the denominator of QA due to the 3^y? I am struggling to conceptualize the fact that QA=QB when the denominator of QB should always be bigger due to the 3^y, thereby making the maximum value of y smaller than the maximum value of x. Thanks!
Quantity A:
Largest x where 200!/7^x is an integer.
So basically, counting factors of 7 in 200!
- Multiples of 7: [200/7] = 28
- Multiples of 7^2 = 49: [200/49] = 4
- Multiples of 7^3 = 343 > 200, so we stop.
Total = 28 + 4 = 32.
So Xmax = 32.
Similary,
Quantity B:
Largest y where 200!/21^y is an integer.
21 = 3 × 7, so we count factors of 3 and 7 in 200! and take the smaller. (as only when both 3 & 7 ae together, is when we can make a 27)
-
Factors of 3:
[200/3] = 66
[200/9] = 22
[200/27] = 7
[200/81] = 2
Total = 97. -
Factors of 7 = 32 (calculated above).
Minimum = min(97,32) = 32.
So Ymax = 32.
Final Answer: Xmax = Ymax = 32 → Quantities are equal. So Option C
I hope you understood the solution!
Yes that makes sense, thank you so much for your help!
No problem