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A number divided by a divisor leaves a remainder of 25. When the number is doubled and divided by the same divisor, the remainder is 11. What is the divisor's value?
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Solution: Step 1: Let the number be N and the divisor be D. Step 2: N = aD + 25, where 'a' is the quotient. Step 3: 2N = 2(aD + 25) = 2aD + 50. Step 4: When 2N is divided by D, the remainder is 11: 2aD + 50 = bD + 11, where 'b' is another quotient. Step 5: Rearrange: 50 - 11 = bD - 2aD, 39 = D(b - 2a). Step 6: Since 39 is a prime number, D must be 39 or 1. Step 7: If D = 1, it doesn't fit typical divisor problems as 1 doesn't produce meaningful remainders in this context. Step 8: Therefore, D = 39.
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What is the remainder when the 100th factorial raised to the power of 100 is divided by 23?
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Solution: Step 1: Recall that for any integer n, if n! is divided by a prime number p, the remainder is 0 if n >= p. Step 2: Since 100! includes 23 as one of its factors (as 23 is a prime number less than 100), 100! is divisible by 23. Step 3: Therefore, (100!)^100 will also be divisible by 23, meaning the remainder is 0.
3
A number divided by 54 gives a remainder of 3. What remainder does it give when divided by 18?
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Solution: Step 1: Express the number as 54k + 3. Step 2: Rewrite 54k as 18(3k) to show its relation to 18. Step 3: Express the number as 18(3k) + 3. Step 4: Conclude that when divided by 18, the remainder is 3.
4
A number n gives a remainder of 3 when divided by 4. What is the remainder when 2n is divided by 4?
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Solution: Step 1: Let n = 4q + 3, where q is an integer. Step 2: Then, 2n = 2(4q + 3) = 8q + 6. Step 3: Rewrite 2n as 4(2q + 1) + 2. Step 4: When 2n is divided by 4, the remainder is 2.
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The numbers from 101 to 150 are concatenated. What is the remainder when this concatenated number is divided by 3?
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Solution: Step 1: Calculate the sum of numbers from 101 to 150 using the formula for summation: Sum = ((first_term + last_term) / 2) * total_terms Step 2: Here, first_term = 101, last_term = 150, total_terms = 50 Step 3: Calculate the sum: Sum = ((101 + 150) / 2) * 50 = 6275 Step 4: Find the digital sum: Digital sum = 6 + 2 + 7 + 5 = 20 Step 5: Further simplify the digital sum: 2 + 0 = 2 Step 6: Calculate the remainder when the digital sum is divided by 3: Remainder = 2 / 3 = 2
6
A gardener has a certain number of shrubs to plant in rows. When trying to plant 8, 12, or 16 shrubs per row, there are always 3 shrubs left. However, when planting 7 shrubs per row, there are none left. What is the total number of shrubs?
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Solution: Step 1: Find LCM of 8, 12, 16 = 48 Step 2: Check numbers of the form 48k + 3 for divisibility by 7 Step 3: For k = 1, 48 + 3 = 51 (not divisible by 7) Step 4: For k = 2, 96 + 3 = 99 (not divisible by 7) Step 5: For k = 3, 144 + 3 = 147 (divisible by 7) Step 6: Required number of shrubs = 147
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