The inconsistency of numbers

Number Theory Level pending

If we know that 19 6 { 19 }^{ 6 } and 23 6 { 23 }^{ 6 } each have 7 factors.

How may factors does 23 6 − 19 6 { 23 }^{ 6 }-{ 19 }^{ 6 } have?


The answer is 96.

Joel Yip by Joel Yip , 21, Singapore

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1 solution

Tom Engelsman
Dec 13, 2016

2 3 6 − 1 9 6 = ( 2 3 3 + 1 9 3 ) ( 2 3 3 − 1 9 3 ) = ( 23 + 19 ) ( 2 3 2 − 23 ⋅ 19 + 1 9 2 ) ( 23 − 19 ) ( 2 3 2 + 23 ⋅ 19 + 1 9 2 ) 23^{6} - 19^{6} = (23^{3} + 19^{3})(23^{3} - 19^{3}) = (23 + 19)(23^2 - 23\cdot19 + 19^2)(23 - 19)(23^2 + 23\cdot19 + 19^2)

or the prime factorization: 42 ⋅ 453 ⋅ 4 ⋅ 1327 = ( 2 ⋅ 3 ⋅ 7 ) ( 3 ⋅ 151 ) ( 2 2 ) ( 1327 ) = 2 3 ⋅ 3 2 ⋅ 7 1 ⋅ 15 1 1 ⋅ 132 7 1 . 42 \cdot 453 \cdot 4 \cdot 1327 = (2\cdot3\cdot7)(3\cdot151)(2^2)(1327) = 2^3 \cdot 3^2 \cdot 7^1 \cdot 151^1 \cdot 1327^1.

Hence, 2 3 6 − 1 9 6 23^6 - 19^6 has 4 ⋅ 3 ⋅ 2 ⋅ 2 ⋅ 2 = 96 4 \cdot 3 \cdot 2 \cdot 2 \cdot 2 = \boxed{96} positive integer factors.

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