Real Numbers — Flashcards
Gujarat Board · Class 10 · Mathematics
24 flashcards for Real Numbers (Gujarat Board Class 10 Mathematics) to test yourself on key terms and facts. Sample: "Find the prime factorization.
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Find the prime factorization of 32760 using a factor tree method
Answer
Step 1: Divide 32760 by 2 → 16380 Step 2: Divide 16380 by 2 → 8190 Step 3: Divide 8190 by 2 → 4095 Step 4: Divide 4095 by 3 → 1365 Step 5: Divide 1365 by 3 → 455 Step 6: Divide 455 by 5 → 91 Step 7: D…
Express 123456789 in prime factor form if 123456789 = 3² × 3803 × 3607. Verify that 3803 and 3607 are prime.
Answer
Prime factorization: 123456789 = 3² × 3803 × 3607 To verify 3803 is prime: - Check divisibility by primes up to √3803 ≈ 61.7 - Test primes: 2 (odd, so no), 3 (3+8+0+3=14, not divisible), 5 (doesn't e…
State the Fundamental Theorem of Arithmetic. Why is it important?
Answer
The Fundamental Theorem of Arithmetic: Every composite number can be expressed (factorised) as a product of primes, and this factorisation is unique, except for the order in which the prime factors oc…
Find the prime factorization of 2520
Answer
Using the division method: Step 1: 2520 ÷ 2 = 1260 Step 2: 1260 ÷ 2 = 630 Step 3: 630 ÷ 2 = 315 Step 4: 315 ÷ 3 = 105 Step 5: 105 ÷ 3 = 35 Step 6: 35 ÷ 5 = 7 Step 7: 7 ÷ 7 = 1 Prime factorization: 25…
Solve: If 2 divides a², then does 2 divide a? Prove using Theorem 1.2
Answer
Yes, if 2 divides a², then 2 must divide a. Here's the proof: Given: 2 divides a² (i.e., a² = 2k for some integer k) By Theorem 1.2: Let p be a prime. If p divides a², then p divides a. Proof reaso…
Prove that √2 is irrational using proof by contradiction
Answer
Proof by contradiction: Assumption: Assume √2 is rational (opposite of what we want to prove) Step 1: If √2 is rational, then √2 = a/b where a and b are coprime integers (no common factors except 1)…
Find the prime factorization of 1771. (Hint: 1771 = 7 × 11 × 23)
Answer
Using the division method: Step 1: Check if divisible by 2 - No (1771 is odd) Step 2: Check if divisible by 3 - 1+7+7+1=16, not divisible by 3 Step 3: Check if divisible by 5 - No (doesn't end in 0 or…
When do you use the Fundamental Theorem of Arithmetic? Provide 3 applications.
Answer
Applications of the Fundamental Theorem of Arithmetic: 1. FINDING HCF/GCD (Highest Common Factor / Greatest Common Divisor) Example: Find HCF of 72 and 120 - 72 = 2³ × 3² - 120 = 2³ × 3 × 5 …
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