week 1 practice problems pending

Day 2:

Q1) What is the three digit number that is equal to 4 times the product of its digits?
S1.
None of the digits can be 0.
c has to be 2,4,6,8
100a + 10b + c = 4abc
=> a = (10b + c)/(4bc - 100)
=> 4bc - 100 > 0 => bc > 25

Now try values of c one by one:
c = 2 => b > 12.5 not possible since b is a single digit
c = 4 => b >= 7
b = 7 => a = 74/12 not integer
b = 8 => a = 84/28 = 3, it works.
b = 8, c = 4 => a = 3
Now check:
384 = 4.3.8.4 = 4.96 = 384.
Answer = 384.

Q2. Show that no square integer ends with 4 ones or 4 nines.
S2.
Square of any odd integer mod 8 = 1.1,3.3,5.5,7.7 = 1
If it ends in 1111 then n = k.10,000 + 1111
k mod 8 = 0 + 111 = 7
So not possible.
Similarly
k.10,000 + 9999 mod 8 = 0 + 999 mod 8 = 7
H.P.

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