Playing with Numbers

Q1: Fill in the blanks:

i. If 84y6 is exactly divisible by 3, then the least value of y is __________.

Step: Sum of digits = 8 + 4 + y + 6 = 18 + y
To be divisible by 3, \(18 + y\) must be divisible by 3.
⇒ \(y = 0\) (least such value)
Answer: y = 0

ii. If 73 m 4 is exactly divisible by 9, then the least value of m is __________.

Step: Sum of digits = 7 + 3 + m + 4 = 14 + m
To be divisible by 9: \(14 + m = 18\) → \(m = 4\)
Answer: m = 4

iii. If m5 n7 is exactly divisible by 3, then the least value of (m + n) is __________.

Step: Sum = m + 5 + n + 7 = m + n + 12
⇒ To be divisible by 3, \(m + n + 12\) must be divisible by 3.
⇒ So, \(m + n\) must be divisible by 3 ⇒ least = 3
Answer: m + n = 3

iv. If x4y5z is exactly divisible by 9, then the least value of (x + y + z) is __________.

Step: Sum = x + 4 + y + 5 + z = x + y + z + 9
⇒ x + y + z must be divisible by 9 ⇒ least = 9
Answer: x + y + z = 9

v. If 7912p is exactly divisible by 8, then the least value of p is __________.

Step: Consider last 3 digits: 12p
Try values: 120 ÷ 8 = 15 (valid) ⇒ \(p = 0\)
Answer: p = 0