2018 AMC 8

Complete problem set with solutions and individual problem pages

Problem 18 Hard

How many positive factors does 23,232 have?

  • A.

    9

  • B.

    12

  • C.

    28

  • D.

    36

  • E.

    42

Answer:E

Solution 1

We can first find the prime factorization of 23,232, which is 2^6\cdot3^1\cdot11^2. Now, we add one to our powers and multiply. Therefore, the answer is (6+1)\cdot(1+1)\cdot(2+1)=7\cdot2\cdot3=\boxed{\textbf{(E) }42}

Note: 23232 is a large number, so we can look for shortcuts to factor it. One way to factor it quickly is to use 3 and 11 divisibility rules to observe that 23232 = 3 \cdot 7744 = 3 \cdot 11 \cdot 704 = 3 \cdot 11^2 \cdot 64 = 3^1 \cdot 11^2 \cdot 2^6.

Another way is to spot the "32" and compute that 23232 = 32\cdot(101 + 10000/16) = 32\cdot (101+ 5^4) = 32\cdot 726 = 32 \cdot 11 \cdot 66.

A third way to factor it is to observe 23232 = 24000 - 768. Factoring out the 3 gives us 3(8000 - 256). Since 8000 = 2^6 \cdot 5^3 and 256 = 2^8, we have 2^6 \cdot 3 (5^3 - 2^2) = 2^6 \cdot 3 (125-4) = 2^6 \cdot 3 \cdot 121 = 2^6 \cdot 3 \cdot 11^2.

 

Solution 2

Observe that 69696 = 264^2, so this is \frac{1}{3} of 264^2 which is 88 \cdot 264 = 11^2 \cdot 8^2 \cdot 3 = 11^2 \cdot 2^6 \cdot 3, which has 3 \cdot 7 \cdot 2 = 42 factors. The answer is \boxed{\textbf{(E) }42}.

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