A student writes positive integers on each of a cube (ABCDEFGH). On each vertex student writes the product of its three adjacent faces. If the integers on faces ABCD, BGFC and CDEF are x1, X, and Xz respectively, then the value at vertex C is X,X2X3. If the sum of the values at all vertices is 154, what is the sum of all integers written on all faces ?
Answers
Given : positive integers written on each face of a cube (ABCDEFGH). On each vertex student writes the product of its three adjacent faces. sum of the values at all vertices is 154
To find : what is the sum of all integers written on all faces
Solution:
Let say integers written on faces
are a , b , c , d , e , f
Sum of Values at vertices
= abc + abe + acd + ade + fbc + fbe + fcd + fde
= (a + f) (bc + be + cd + de)
= (a + f) (b(c + e) + d(c + e))
= (a + f) (b + d)(c + e)
(a + f) (b + d)(c + e) = 154
=> (a + f) (b + d)(c + e) = 2 * 7 * 11
( 1 can not be factor as sum of two integer would be minimum 2 ) as here 0 can not be used
a + b + c + d + e + f = 2 + 7 + 11
=> a + b + c + d + e + f = 20
sum of all integers written on all faces= 20
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