Competition · AMC preparation · step 4 of 4
AMC 10 · 2002B · #14
Grade 8 number-theoryPick an answer.
AMC 10 2002 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
The number is astronomically large, so Tool #9 (Solve an Easier Related Problem) replaces the impossible arithmetic with an easier one: break 25 and 64 into powers of the primes 5 and 2, where taking a square root just means halving the exponents. Tool #15 (Organize Information in More Ways) then regroups the leftover 2s and 5s so that equal numbers of them pair into 10s. Tool #5 (Look for a Pattern) finishes it: multiplying by a power of 10 only tacks zeros onto the end, so the digit sum comes from the few nonzero digits in front.
Break the bases into primes
Since 25=5² and 64=2⁶, multiplying exponents gives 25⁶⁴ · 64²⁵=5¹²⁸ · 2¹⁵⁰ — the same number, purely in primes.
A power raised to a power just multiplies the exponents, turning messy bases into clean prime powers.
8.EE.A.1Solve An Easier Related ProblemTake the root by halving exponents
A square root halves each exponent, and both exponents are even, so N=5⁶⁴ · 2⁷⁵.
Squaring doubles an exponent, so undoing it — the square root — cuts the exponent in half.
8.EE.A.1Solve An Easier Related ProblemPair twos and fives into tens
Split 2⁷⁵=2⁶⁴ · 2¹¹ and pair 2⁶⁴ with 5⁶⁴ into (2 · 5)⁶⁴=10⁶⁴, leaving N=2¹¹ · 10⁶⁴.
Every 2 paired with a 5 becomes a 10, and 10s are exactly what create trailing zeros.
Every two paired with a five becomes a ten, and tens are exactly what create trailing zeros.
▸ Why?
Every number has one prime recipe, so the supply of twos and fives is fixed in advance.
▸ Why?
Multiplying by ten shifts every digit one place left and drops a zero in behind it.
Evaluate the front number
Only the leftover 2¹¹ carries real digits: 2¹¹=2048, so N=2048 · 10⁶⁴.
The paired-off 10s only add zeros, so the meaningful digits all live in the small factor out front.
6.EE.A.1Solve An Easier Related ProblemAppend the zeros and add the digits
N is 2048 followed by 64 zeros, and zeros add nothing, so the digit sum is 2+0+4+8=14 — choice (B).
Multiplying by a power of 10 only slides digits left and fills in zeros, which never change the digit sum.
5.NBT.A.2Look For A PatternTurn scary bases into powers of 2 and 5, pair every 2 with a 5 to make 10s (which are just trailing zeros), and the tiny leftover 2¹¹=2048 hands you the digit sum 2+0+4+8=14.
- Break the bases into primes
- Take the root by halving exponents
- Pair twos and fives into tens
- Evaluate the front number
- Append the zeros and add the digits
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