Math Reasoning Exercise: Set 6 - Questions 1 to 3
Question 1
17.3.24 is a Sunday.. Which day of the week is 17.12.24
Concept: Number patterns and periodicity.
This is a cool concept to learn and apply.
If you observe the days of the month, they repeat year after year. So its possible to predict the day, if we have the date.
Lets understand the concept.
If every month of the year had only 28 days. Then there will be no change in the day.. as every week has 7 days and the pattern repeats.
Mon Tue Wed Thur Fri Sat Sun Mon Tue Wed Thur Fri Sat Sun…
So if 2nd Jan is Tuesday then 7 days days later 9th Jan will be Tuesday or 14 days later.. 16th Jan will be Tuesday. Or 35 days later.. (31 days of Jan + 4 days of Feb = 4th Feb) will be a Tuesday.
7/14/35 are all multiples of 7. Pattern repeats every 7.
But some months have 28 days others have 30 and 31 days.. so lets calculate the extra days.
Every month in a year has 28 days. ( I have chosen 28 as 28 is a multiple of 7 as pattern repeats after 7)
Apart from the 28 days.. some months have extra days.. lets calculate.
January, March, May, July, August, October, December have extra 3 days.
April, June, Sept, Novemeber have extra 2 days.
February has 0 extra days for a non leap year and 1 extra day for leap year(year should be divisible by 4)
So if 17th march is Sunday. Lets calculate
Step 1: given day to end of the month: 18th march to 31st march = 14 days = (31-18+1)
Step 2: Lets calculate extra days for the subsequent months
April -2
may – 3
June -2
July -3
Aug-3
Sept-2
Oct–3
Nov-2
Step 3: Calculate the days in the last month – Dec 1 to 16days. =16. We are calculating till 16th December.
Lets add 14 +2+3+2+3+3+2+3+2+16= Total = 50 days
50 days = 49 days + 1= 7×7+1
Remember its patterns of 7
49 is a multiple of 7 so we have to focus on the extra day.i.e 1
17th march is Sunday. So lets calulate 1 more day.
16th December is Sunday + 1 = Monday
Hence 17th December is Tuesday
Now you calculate
Practice sums
17.3.24 is a Sunday.. Which day of the week is
1. 18th June 2024
2. 20th Oct 2024
3. 25 November 2024
Answers
1. Tuesday
2. Sunday
3. Monday.
Question 2
Concept – prime factorization
Let’s do prime factorization of all the numbers, other than the primes
4= 22
6= 2×3
8 = 23
9= 32
10=2×5
12 = 4×3 =22 x3
14 = 7×2
15 = 5×3
Now lets multiply
1x2x3x22x5x(2×3)x7x23x32x(2×5)x11x(22 x3)x13x(2×7)x(5×3)
So we can see all the prime numbers from 2 to 11
We need to observe 5 and 2
One 5 x one 2 = 10 (one zero)
Two 5 x two 2 = 5×5 x2 x 2 = 100 (two zeros)
Three 5 x three 2 = 5x5x5x2x2x2 = 1000 (three zeros)
So if we calculate the number of 5 and multiply with equal number of 2’s then we can get a multiple of 10.
10 x any number ends in a 0.
100 x any number ends in a 00
So we can see there are three 5’s and three 2’s.
Hence we can group 1x2x3x22x5x(2×3)x7x23x32x(2×5)x11x(22 x3)x13x(2×7)x(5×3) this way
Group 1- 1x3x22x3x7x23x32x11x(22 x3)x13x(2×7)x3
Group 2- 5 x 5 x 5x 2x2x2 ( I have taken out the three 5’s and three 2’s)
Product =group 1 x group 2
=group 1x 5 x 5 x 5x 2x2x2
=group 1 x 1000
The product should end in 000. Hence the second last digit is 0.
Now try another sum
How many zero’s are there in the product of 1x2x3x4x5x6x7x8……..23x24x25?
Answer.
6 zeros. as there six 5’s