( 10a + 11b + c = k^2 ). Rearrange: ( c = k^2 - 10a - 11b ).
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A number with exactly 5 divisors must be of the form (p^4) where (p) is prime (since divisor count = exponent+1, so exponent=4). (p^4 < 100) → (p^4 < 100). (2^4=16), (3^4=81), (5^4=625) (too big). So (n = 16) and (81). That’s 2 numbers. ( 10a + 11b + c = k^2 )
The National Sprint Round is designed to be the ultimate test of speed and accuracy for middle schoolers. MATHCOUNTS Foundation : 30 short-answer problems to be solved in 40 minutes. Calculators : Strictly not permitted Difficulty Curve (p^4 < 100) → (p^4 < 100)
To excel in the National Sprint Round, top competitors employ specific tactical approaches:
So for S where 7S ≡ 0 mod 9 → 7S multiple of 9 → since gcd(7,9)=1, S multiple of 9. S=9,18. For S=9: C=0 or 9 (2 values). For S=18: C=0 or 9 (2 values). All other S: 1 value.