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题库 / Prep2007E2-DS-129
If t is a positive integer and r is the remainder when ~$t^2 + 5t + 6$~ is divided by 7, what is the value of r ?
(1) When t is divided by 7, the remainder is 6.
(2) When t2 is divided by 7, the remainder is 1.
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
EACH statement ALONE is sufficient.
Statements (1) and (2) TOGETHER are NOT sufficient.
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(1) When t is divided by 7, the remainder is 6 --> t=7q+6t=7q+6 --> (t+2)(t+3)=(7q+8)(7q+9)(t+2)(t+3)=(7q+8)(7q+9). Now, no need to expand and multiply all the terms, just notice that when we expand all terms but the last one, which will be 8*9=72, will have 7 as a factor and 72 yields the remainder of 2 upon division by 7. Sufficient.
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