The Quest for 700: Weekly GMAT Challenge (Answer)

The hardest part about this problem is understanding how p(n) is defined. After reading the definition, focus on the example given. We are told that p(102) = 20, since 20 = 5¹3⁰2². Study this example: the 102 shows up in the exponents of the primes: 5¹3⁰2².

Try constructing another example or two. Use small numbers with just units and tens digits, since you know explicitly what to do with units and tens digits. For instance, p(11) is equal to 3¹2¹ = 6. Likewise, p(12) is equal to 3¹2² = 12, in fact.

Now we are asked for the smallest integer that is NOT equal to p(n) for any permissible n. In other words, you cannot construct the prime factorization of this number using the process above.

We should now think of limitations on the results. Since what we’re doing is putting the digits of n into separate exponents, the highest that any particular exponent can go is 9 (the largest digit in the base-10 system). This means that 2¹⁰ is not constructible, since only the units digit of n goes in as the exponent of 2.

Since 2 is the smallest prime, 2¹⁰ is the smallest number that you can’t reach with this process. 2¹⁰ = 1,024.

Notice that you can in fact output 1 as p(n) if you put in n = 0, since 2⁰ = 1.