QUESTION 3
Easy
How many three-digit numbers are divisible by 7?
(Hint: All three-digit numbers divisible by 7 form an AP. Find the smallest and largest such three-digit numbers.)
SOLUTION
1
Find the smallest three-digit multiple of 7
100 ÷ 7 = 14 remainder 2
So smallest multiple: 15 × 7 = 105
So smallest multiple: 15 × 7 = 105
2
Find the largest three-digit multiple of 7
999 ÷ 7 = 142 remainder 5
So largest multiple: 142 × 7 = 994
So largest multiple: 142 × 7 = 994
3
Form the AP
The AP is 105, 112, 119, ..., 994
4
Find the number of terms
\(l = a + (n − 1)d\)
⇒ \(994 = 105 + (n − 1)7\)
⇒ \(889 = 7(n − 1)\)
⇒ \(127 = n − 1\)
⇒ \(n = 128\)
⇒ \(994 = 105 + (n − 1)7\)
⇒ \(889 = 7(n − 1)\)
⇒ \(127 = n − 1\)
⇒ \(n = 128\)
🏆
Final Answer : Number of three-digit multiples of 7 = 128
Concept Note
Multiples of a number form an AP.
To find the number of terms, use
\(n = \frac{l − a}{d} + 1\),
where \(a = \)first term, \(l = \) last term, \(d = \) common difference