A student opened her math book and computed the sum of the numbers on two facing pages. Then she turned to the next page and computed the sum of the numbers on these two facing pages. Finally, she computed the product of the two sums, and her calculator displayed the number 85,845. What were the four page numbers?
Let the page numbers be represented by a, b, a+1 (next page after a), and b+1 (next page after b).
Since the sum of the numbers on pages a and b is computed, then we can write:
a + b = sum1
Similarly, the sum of the numbers on pages a+1 and b+1 is computed, then we can write:
(a+1) + (b+1) = sum2
a + b + 2 = sum2
Given that the product of the two sums is 85,845, we can write:
sum1 * sum2 = 85,845
(a + b)(a + b + 2) = 85,845
a(a + b + 2) + b(a + b + 2) = 85,845
a^2 + 2a + ab + b^2 + 2b = 85,845
Substitute sum1 and a+b into the equation:
a^2 + 2a + ab + b^2 + 2b = 85,845
a^2 + 2a + ab + b^2 + 2b = 85,845
Since there are various numbers that could satisfy this equation, let's test some combinations to find the page numbers that work.
Page numbers: 121, 122, 123, 124
121 + 122 = 243
123 + 124 = 247
243 * 247 = 60,021 which is not equal to 85,845
Let's try another combination.
Page numbers: 134, 135, 136, 137
134 + 135 = 269
136 + 137 = 273
269 * 273 = 73,437 which is not equal to 85,845
Let's try another combination.
Page numbers: 185, 186, 187, 188
185 + 186 = 371
187 + 188 = 375
371 * 375 = 139,125 which is not equal to 85,845
Let's try another combination:
Page numbers: 194, 195, 196, 197
194 + 195 = 389
196 + 197 = 393
389 * 393 = 153,177 which is not equal to 85,845
Let's try another combination:
Page numbers: 126, 127, 128, 129
126 + 127 = 253
128 + 129 = 257
253 * 257 = 64,921 which is not equal to 85,845
It seems that no combination of page numbers satisfies the given condition, which means that there may be a mistake in the calculations.