F.4 maths @_@ 20marks...

(1) It is given that N is a positive integer. Prove that 2[5^(N+1)] is a multiple of 10.
2 x 5N+1 = 2 x ( 5N x 5 ) = 2 x 5 x 5N = 10 x 5N
Since N is a positive integer,
5N is also a positive integer.
Therefore 2 x 5N+1 = 10a , where a is an integer.
2 x 5N+1 is a multiple of 10.

(2) The tens digit and the units digit of a two-digit number are a and b respectively.
a) Express the value of the two-digit number in terms of a and b.
The value of tens digit is : 10a
The value of units digit is : b
The two-digit number is : 10a + b

b) By interchanging the tens digit and units digit of the number, we can obtain a new two-digit number. Prove that the sum of these two-digit numbers is divisible by 11.
The new two-digit number is : 10b + a
The sum of these two-digit numbers is :
( 10a + b ) + ( 10b + a )
= 11a +11b
= 11 x ( a + b )
= 11c where c is an integer.
The sum of these two-digit numbers is divisible by 11.