Optional enrichment: Perfect powers, divisibility and constrained prime factors
Example 1
N=2x×3y, where x and y are positive integers and x=y+2. N has 35 positive divisors. Find x.
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Answer: 6
- (x+1)(y+1)=35.
- Substitute x=y+2:(y+3)(y+1)=35.
- The positive solution is y=4
- x=6.
Example 2
N=213×313. How many positive divisors of N are divisible by 6 but are not perfect squares?
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Answer: 133
- Divisibility by 6 gives 13×13 choices of positive exponents.
- Among these, square divisors have 6 choices of positive even exponent for 2 and 6 for 3.
- Subtract 36 from 169.
Example 3
N=2x×3y, where x and y are positive integers and x=y+2. N has 48 positive divisors. Find x.
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Answer: 7
- (x+1)(y+1)=48.
- Substitute x=y+2:(y+3)(y+1)=48.
- The positive solution is y=5
- x=7.