3. The sum of digits of a 2-digit number is 3. If the number is 6 times the ones digit, find the number.
4. The sum of digits of a 2-digit number is 9. If the digits are reversed, the new number is 27 more
than the original number. Find the original number.
Answers
Answer:
ANSWER
Let the tens and the units digits in the number be x and y, respectively.
So, the number may be written as 10x+y.
According to the given condition.
10x+y=6(x+y)+3⇒4x−5y=3 .....(i)
If we interchange the digits of Original number then we get new number. i.e 10y+x
According to the given condition
10y+x+18=10x+y
⇒9x−9y=18⇒x−y=2 .....(ii)
Now multiplying equation (ii) by 4. we get,
4x−4y=8 ....(iii)
On subtracting (iii) from (i). we get,
4x−5y−(4x−4y)=3−8
∴y=5
On substituting y=5 in (ii). we get,
x−5=2⇒x=7
∴ number is 10x+y=10(7)+5=75
4)There is a simple test for divisibility by 9 that states a number is divisible by 9 if (and only if) the (repeated) digit sum of the number is divisible by 9. For example, 18 and 81 are divisible by 9 because 1+8=9.
So the two-digit number we are looking for is divisible by 9.
Moreover the difference to another number divisible by 9 is 27. For the majority of numbers (those ending in 4 and higher), adding 27 lowers the units place by 3 so I started thinking about those, and quickly found 36. But a more structured way is nothing that the two-digit multiples of nine pair up:
18 <=> 81
27 <=> 72
36 <=> 63
…
And of course, the difference is a multiple of 9 too, decreasing in steps of 2 because you increase one and decrease the other number by 9 each time:
81 - 18 = 9 x 9 - 2 x 9 = 7 x 9
72 - 27 = (9–1) x 9 - (2+1) x 9 = 5 x 9
63 - 36 = 3 x 9
54 - 45 = 1 x 9
…
Now of course, it's no coincidence that 27 = 3 x 9 so starting from either side of the above list quickly finds (36, 63) without solving any equations!
Hope it helps u dear