CAT — Classification of Numbers
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In a 3-digit number N, the digits are non-zero and distinct such that none of the digits is a perfect square, and only one of the digits is a prime number. Then, the number of factors of the minimum possible value of N is
Non-zero digits excluding perfect squares (1, 4, 9): allowed digits are 2, 3, 5, 6, 7, 8. Among these, prime digits are 2, 3, 5, 7 and composite digits are 6, 8. Since only one digit is prime, the number must contain both composites (6 and 8) and exactly one prime.
To minimise N: hundreds digit = 6, tens digit = 2 (smallest prime), units digit = 8. So N = 628.
Prime factorisation: 628 = 2² × 157 (157 is prime). Number of factors = (2+1)(1+1) = 6.
The number of positive integers less than 50, having exactly two distinct factors other than 1 and itself, is
Step 1: Understand the condition A number has exactly two distinct factors other than 1 and itself if it has exactly 4 divisors in total. Numbers with exactly 4 divisors are of two types: p³ where p is prime, or p × q where p and q are distinct primes. Step 2: Find numbers of the form p³ less than 50 2³ = 8, 3³ = 27 → 2 such numbers. Step 3: Find numbers of the form p × q less than 50 Using prime 2: 6, 10, 14, 22, 26, 34, 38, 46 Using prime 3: 15, 21, 33, 39 Using prime 5: 35 Using prime 7: 7 × 11 = 77 > 50, stop. Total: 13 such numbers. Step 4: Total count = 2 + 13 = 15
Brishti went on an 8-hour trip in a car. Before the trip, the car had travelled a total of x km till then, where x is a whole number and is palindromic, i.e., x remains unchanged when its digits are reversed. At the end of the trip, the car had travelled a total of 26862 km till then, this number again being palindromic. If Brishti never drove at more than 110 km/h, then the greatest possible average speed at which she drove during the trip, in km/h, was
110
90
100
80
100
Let the odometer reading before the trip be x km.
Given,
● Final reading = 26862 km (a palindrome)
● Trip duration = 8 hours
● Maximum speed at any time = 110 km/h
Hence, the maximum possible distance travelled is 110 × 8 = 880 km
Therefore, 26862 − x ≤ 880 → x ≥ 26862 − 880 → x ≥ 25982
To maximize the average speed, we must maximize the distance travelled.
So, we need the smallest palindromic number greater than or equal to 25982.
Check palindromes:
● Mirror 259 → 25952 (less than 25982, so not possible)
● Next palindrome is 26062
Thus, x = 26062
Distance travelled = 26862 − 26062 = 800 km
Average speed = 800 ÷ 8 = 100 km/h
Answer:
C. 100
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