Euclid's lemma, fundamental theorem, irrationals, HCF/LCM MCQs

Q1. How many positive integers from 1 to 1000 are divisible by 2 or 5?

• A.600
• B.500
• C.700
• D.550

Answer: A. 600

Spot-the-setup - a typical math numerical. The data on the table: 1, 1000, 2, 5. We are after How many positive integers from 1 to 1000 are divisible by 2 or 5?. Tool of choice - Inclusion-exclusion: ⌊1000/2⌋ + ⌊1000/5⌋ − ⌊1000/10⌋ = 500 + 200 − 100 = 600. Rearrange it for the unknown before substituting. Numbers in: Inclusion-exclusion: ⌊1000/2⌋ + ⌊1000/5⌋ − ⌊1000/10⌋ = 500 + 200 − 100 = 600. Lock in option A) 600.

Q2. The decimal expansion of 17/3125 terminates after how many places?

• A.5
• B.4
• C.3
• D.Does not terminate

Answer: A. 5

Given: 17; 3125. To find: how many places?. The relation that links these is - 3125 = 5⁵, of the form 2⁰·5⁵, so it terminates after max(0,5)=5 places. This is the equation that links the given quantities to the unknown (math, chapter 'Real Numbers'). Substituting: 3125 = 5⁵, of the form 2⁰·5⁵, so it terminates after max(0,5)=5 places → 17/3125 = 17·32/10⁵ = 544/100000 = 0.00544. Thus the answer is A) 5.

Q3. Which of the following has a non-terminating recurring decimal?

• A.23/(2³·5²·3)
• B.13/3125
• C.7/8
• D.23/200

Answer: A. 23/(2³·5²·3)

Key chapter idea: A reduced rational p/q has terminating decimal iff q is of the form 2^a·5^b. 23/(2³·5²·3) has factor 3 in the denominator, so it recurs. So the correct option is A) 23/(2³·5²·3).