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# Combinatorics: Factorials- Basic Definitions

If $$t$$ is a non-negative integer, is $$21!$$ divisible by $$3^t$$? >(1) $$t$$ is the product of two distinct single-digit prime numbers that are each less than $$7$$. >(2) $$0 \lt t \lt 9$$
Correct. [[snippet]] Stat. (1) refers to the prime numbers smaller than 7, which are 2, 3, and 5. Thus, $$t$$ is between $$2 \times 3 = 6$$ and $$3 \times 5 = 15$$, inclusive, so $$t$$ may be smaller than, larger than, or equal to 9. Therefore, **Stat.(1) → Maybe → IS → BCE**. Stat. (2) states that $$t$$ is between 0 and 9, exclusive. Thus, whatever the value of $$t$$, it is smaller than 9, so we have enough values of 3 in $$21!$$ to reduce all the 3's in $$3^t$$, and the answer is a definite "Yes". **Stat.(2) → Yes → S → B**.
Incorrect. [[snippet]] Stat. (1) refers to the prime numbers smaller than 7, which are 2, 3, and 5. Thus, $$t$$ is between $$2 \times 3 = 6$$ and $$3 \times 5 = 15$$, inclusive, so $$t$$ may be smaller than, larger than, or equal to 9. Therefore, **Stat.(1) → Maybe → IS → BCE**.
Incorrect. [[snippet]] Stat. (1) refers to the prime numbers smaller than 7, which are 2, 3, and 5. Thus, $$t$$ is between $$2 \times 3 = 6$$ and $$3 \times 5 = 15$$, inclusive, so $$t$$ may be smaller than, larger than, or equal to 9. Therefore, **Stat.(1) → Maybe → IS → BCE**. What about stat. (2) alone? Does it provide sufficient data to determine whether $$t \le 9$$?
Incorrect. [[snippet]] Stat. (1) refers to the prime numbers smaller than 7, which are 2, 3, and 5. Thus, $$t$$ is between $$2 \times 3 = 6$$ and $$3 \times 5 = 15$$, inclusive, so $$t$$ may be smaller than, larger than, or equal to 9. Therefore, **Stat.(1) → Maybe → IS → BCE**.
Incorrect. The question stem states that $$t$$ is an integer. Does that put the statements in a different light?
Ok. [[snippet]] Stat. (2) states that $$t$$ is between 0 and 9, exclusive. Thus, whatever the value of $$t$$, it is smaller than 9, so we have enough values of 3 in $$21!$$ to reduce all the 3's in $$3^t$$, and the answer is a definite "Yes". **Stat.(2) → Yes → S → BD**.
Just a bit more care . . .
Statement (1) ALONE is sufficient, but Statement (2) alone is not sufficient to answer the question asked.
Statement (2) ALONE is sufficient, but Statement (1) alone is not sufficient to answer the question asked.
BOTH Statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.
I remembered that $$t$$ is an integer.