m and n are both prime numbers. Find the values of m and n so that 784 × m/n is a perfect cube.

Steps to Solve

1. Factor the number 784

To begin solving for ( m ) and ( n ), we need to factor 784 into its prime factors.

Starting with ( 784 ):

• Divide by 2: ( 784 \div 2 = 392 )
• Divide by 2: ( 392 \div 2 = 196 )
• Divide by 2: ( 196 \div 2 = 98 )
• Divide by 2: ( 98 \div 2 = 49 )
• Divide by 7: ( 49 \div 7 = 7 )
• Divide by 7: ( 7 \div 7 = 1 )

Thus, the prime factorization of ( 784 ) is:

$$ 784 = 2^4 \times 7^2 $$

2. Determine the exponents for a perfect cube

For a product to be a perfect cube, all the exponents in its prime factorization must be multiples of 3.

From ( 784 ):

• The exponent of ( 2 ) is ( 4 ).
• The exponent of ( 7 ) is ( 2 ).

3. Adjust the exponents using ( m ) and ( n )

Let ( m = 2^a \times 7^b ) and ( n = 2^c \times 7^d ) where ( a, b, c, d ) are non-negative integers. The modified expression becomes:

$$ 784 \times \frac{m}{n} = 2^{4 + a - c} \times 7^{2 + b - d} $$

To satisfy the perfect cube condition:

• ( 4 + a - c ) needs to be a multiple of ( 3 ).
• ( 2 + b - d ) needs to be a multiple of ( 3 ).

4. Solve for values of ( m ) and ( n )

To make ( 4 + a - c ) a multiple of ( 3 ):

• The possible values are ( 4 + 0, 4 + 1, 4 + 2 ) modulo ( 3 ), leading to:
  • ( 4 ) gives ( 1 ) (mod 3)
  • ( 5 ) gives ( 2 ) (mod 3)
  • ( 6 ) gives ( 0 ) (mod 3)

This means ( 1 - c \equiv 0 ) (mod 3) implies ( c \equiv 1 ) (mod 3) → possible values for ( c: 1, 4, 7 ) which leads us to pick ( c = 1 ) (to be prime). Thus, ( a = 2 ).

To solve for ( 2 + b - d ):

• Set ( d = 1 ) and ( b = 1 ): $$ 2 + b - d = 2 + 1 - 1 = 2 $$ (not a multiple of 3) Keep adjusting: If ( d = 2 ) and ( b = 1 ): $$ 2 + 1 - 2 = 1 $$ (not a multiple of 3) Let ( d = 0 ) and ( b = 1 ): $$ 2 + 1 - 0 = 3 $$ (multiple of 3)

Thus possible values: Let ( m = 2^2 \times 7^1 = 28 ) and ( n = 2^1 = 2 ).

5. Final composition of ( m ) and ( n )

In conclusion, ( m = 7 ) and ( n = 2 ) is indeed valid.

Thus, we find:

• ( m = 7 )
• ( n = 2 ).