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1. find the number of numbers between 100 to 400 which are divisble by either 2,3,5,or 7.
2.find th enumber of numbers between 300 to 400 which are not divisble by 2,3 ,5 and 7.
find the last digit in the expression( 36472)123! x (3476)76! .
3.how can we determine that m! ,where m is any natural number, is of which form wether 4n or 4n 1 or 4n 2 or 4n 3.
4.find the unit digit in each of following expression
a. 22x44x66x88
b.1x22x33x44x......x100100
c. 37123x43144x57266x32157x525!
Answers (1)
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1) First find out the numbers between 100 and 400 which are not divisible by 2. They would be simply equal to half of the total numbers between 100 and 400. Next, find out the numbers divisible by 3 and eliminate out of it even numbers which have already been counted earlier. Follow suit with 5 and 7. Now subtract these numbers from the total and get the answer.
2) This one is simple. Look at ( 36472)123!
Patterns in unit's digits of the powers of 2 is 2, 4, 8, 6 i.e it repeats after 4 terms. Now we have to see the remainder when 123! is divided by 4. Remainder would be zero because many terms in 123! are divisible by 4. Hence units digit of ( 36472)123!
will be 6.
Now look at (3476)76!
Any power of 6 will have units digit = 6. Hence the unit's digit of this number is also 6.
Hence the unit's digit of ( 36472)123! x (3476)76! will be equal to 6.
3) a) 22x44x66x88
units digit of 22 = 4
units digit of 44 = 6
units digit of 66 = 6
units digit of 88 = 6
Thius unit's digit of given product = units digit of 4x6x6x6 = 4
b) Units digit of 1x22x33x44x......x100100 is equal to zero as the product contains powers of 100
c) Units digit of 37123x43144x57266x32157x525! is also equal to zero as 525! contains terms like 10, 100, 200, 500 etc