Reliance
To determine how many consecutive zeros the product of S will end with, we need to find the highest power of 10 that divides the product. This is equivalent to finding the highest power of 5 that divides the product, since the number of factors of 2 will always be greater than the number of factors of 5.
The primes in S are {2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97}.
There are 24 primes in S, so the product of S is:
2 x 3 x 5 x 7 x 11 x 13 x 17 x 19 x 23 x 29 x 31 x 37 x 41 x 43 x 47 x 53 x 59 x 61 x 67 x 71 x 73 x 79 x 83 x 89 x 97
We need to find the highest power of 5 that divides this product. To do this, we count the number of factors of 5 in the prime factorization of each number in S.
5 appears once: 5
5 appears once: 25
5 appears once: 35
5 appears once: 55
5 appears once: 65
5 appears once: 85
So, there are six factors of 5 in the product of S. However, we also need to consider the powers of 5 that arise from the factors 25, 35, 55, and 65.
25 = 5 x 5 appears once: 25
35 = 5 x 7 appears once: 35
55 = 5 x 11 appears once: 55
65 = 5 x 13 appears once: 65
Each of these numbers contributes an additional factor of 5 to the product of S. Therefore, there are 6 + 4 = 10 factors of 5 in the product of S.
Since each factor of 5 corresponds to a factor of 10, we know that the product of S will end with 10 zeros. Therefore, the product of S will end with 10 consecutive zeros
c
Statements :
All mangoes are golden in colour. No golden coloured things are cheap.
Conclusions :
I. All mangoes are cheap.
II. Golden coloured mangoes are not cheap.
125, 106, 88, 76, 65, 58, 53
88
90 inches
Slab = 6(x) where x is each step
Slab = 5(x + 3)
6(x) = 5(x + 3)
6(x) = 5(x) + 15
x = 15 inches
So 6(x) = 6(15) = 90
Check: 5( x + 3) = 5(18) = 90
Slab size is 90 inches
time (dist covrd by husband) (dist covrd by wife)
0.5hr 20 miles 0 miles
1 hr 40 miles 25 miles
1.5 hr 60 miles 50 miles
2 hrs 80 miles 75 miles
2.5 hrs 100 miles 100 miles
Therefore wife catches up after 2.5 hrs..
7 1/2
the answer is 45 km/h
Define x:
Let the total distance be x.
Find the time needed in terms of x for speed at 30 km/h:
Find the time needed in term of x for 40 km./h:
Find the difference in time:
Solve x:
Find the total time needed to for the whole journey:
Given that he was 40 mins late:
Find the minimum speed needed:
Answer: The minimum speed is 45 km/h
7days
224
( b ) P