# Question Bank - Number Theory - Shashank Prabhu, CAT 100 Percentiler

• 500! (1 + something that ends in 0)
So, we have to find the highest power of 5 in 500! which will be 100 + 20 + 4 = 124. Hence, n can take values from 0 to 124 (remember that n is a whole number and the expression will be divisible by 5^0 (i.e. 1) - TRAP!). 125 values in total.

• Q100) A 4-digit number of the form aabb is a perfect square. What is the value of a + b?

• The perfect squares that have the last two digits in the form of bb will end in either 00 or 44. So, we need to consider 38^2, 62^2 and 88^2. Only 88^2 will satisfy the condition and will be equal to 7744 (remember! if you don’t already). So, a + b = 11.

• @shashank_prabhu
@zabeer Sir what should be the correct approach

• @shashank_prabhu

a.........

• @shashank_prabhu
c.................

• @shashank_prabhu 1......................

• @shashank_prabhu is it 72

• 1000 = 2^3 * 5^3
Set A consist of all natural numbers between 1 and 100 which are divisible by 3 or 4 (not by both).
To find the highest power of 1000, we need to see how many 5^3 (highest prime in the factored form) we can get in the product of numbers in set A
3 * 1 * 5,
3 * 2 * 5,
3 * 3 * 5,
3 * 4 * 5, (divisible by both 3 and 4)
3 * 5 * 5,
3 * 6 * 5 (we can stop here as 3 * 7 * 5 is greater than 99 hence not a member of set A)
4 * 1 * 5,
4 * 2 * 5,
4 * 3 * 5, (divisible by both 3 and 4)
4 * 4 * 5 (we can stop here as 4 * 5 * 5 is greater than 99 and hence not a member of set A)

multiply all numbers and we get 9 fives.
So [9/3] = 3 should be the highest power of 1000 in the product of all numbers in set A.

not sure if this is the best method though. still should not take much time right?

@Amar-Rajput can you also share your detailed approach while posting answers? this should help other aspirants too. Cheers!

• @shashank_prabhu @zabeer sure sir will post my solution too , thanks for ur approach

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