Problem 7 · Filter square roots using remainders · 5 points
Made by AI and not reviewed by a real person.
How many four-digit perfect squares leave a remainder of 1 when divided by every whole number from 2 through 7?
- 13
- 1
- 2
- 3
- 8
Show a hint
- Start with the square roots. What can you learn about their final digits from the squares’ final digits?
- After checking division by 3, use (root − 1) × (root + 1) to check division by 7.
Show the answer and solution
Answer: C — 2
A four-digit square has a root from 32 to 99. The square must be odd and leave 1 when divided by 5, so it ends in 1; its root must therefore end in 1 or 9. The possible roots are 39, 41, 49, 51, 59, 61, 69, 71, 79, 81, 89, 91, and 99. Removing roots divisible by 3 leaves 41, 49, 59, 61, 71, 79, 89, and 91. Because root² − 1 = (root − 1)(root + 1), the test for 7 leaves only 41 and 71. Their squares are 1681 and 5041; subtracting 1 gives 1680 and 5040, both divisible by every number from 2 through 7. The answer is 2. The tempting answer 8 forgets the test for 7.