763*312, which number should the * be replaced to make the number divisible by 9?
For a number to be divisible by 9, its digits sum should be divisible by 9.
∴ 7 + 6 + 3+*+ 3+1+2 = 22 +*.
∴ 5 should be written on place of * to make the number divisible by 9.
76215*, the replacement of * by a number gives a number which is divisible by 11, the number will be :
⇒ Sum of digits at odd places
= * + 1 + 6 = * +7
Sum of digits at even places
= 5 + 2 + 7
= 14
For divisibility with 11, *+ 7 = 14
⇒ * = 7
, the values of A, B, C are digits from 1 to 9. What will be value of B?
3c = x B, where, x is the carry,
∴ x + 3B = y B, where y is the carry.
∴ y + 3A = B.
∴ C = 8, B = 4, A = 1.
What will be the sum of first 22 natural numbers, which, are even?
Sum of first 22 even natural numbers
= (2 + 4 + …… + 44)
= 2 (1 + 2 + …… + 22)
= 2 × 11 × 23
= 22 × 23
= 506
One candle was guaranteed to burn for 6 hours, the other for 2 hours. They were both lit at same time. After some time one candle was twice as long as the other. For how long had they been burning?
Let the candles have burnt for ‘x’ hours,
⇒
Which is a 3digit numbers, such that all its digits are prime and the 3 digits are the factors of the number?
735 is a number having all its digits, a prime number and all the digits of 735 are the factors of 735.
Complete the square given below, and find the value of the sum of missing numbers. The sum of the magic square is 34.
We have to find, the value of
(a+b+c+d +e)+(x+ y + z)+ f = x
∴ Sum of numbers of any row/column = 34
∴ a +b+13+6= x+ y + z +13
= e+b +7 +9=c + d +6+ f
= 5 + 16 + a + 2 = x +e+d + 5
= 16 + y + 7 +c= 2 + z + 9 + f = 34
⇒ 2(a+b+c+d +e+ f + x+ y+ z)
+19 + 13 + 16 + 6 + 23+ 5
+23 + 11 34 x 8
⇒ 2x + 116 = 272
⇒ x = 78
Three numbers are such that their sum is 10 and their product is maximum. The product will be :
Let the natural numbers be, y and z.
⇒
⇒
⇒ xyz ≤ 37.03
∴ The limiting value of xyz is 37.03,
∴ The numbers x,y and z are natural.
∴ 37.03 cannot be obtained as a product.
∴ 37 is a prime number.
∴ 36 is the greatest number which can be obtained as a product of 3 natural numbers whose sum is 10.
What will be the one’s place digit of 6^{222}?
60 = 1
6^{1} = 6
6^{2} = 36
6^{3} = 216
6^{4} = 1296
6^{5} = 7776
∴ It is observed that 6^{n} has 6 at its units place.
∴ 6^{222} has 6 as its unit’s place digit.
Find the smallest number which can be expressed as the sum of two cubes of natural numbers.
(a) 1729 = (12)^{3} + (1)^{3}= (10)^{3} + (1)^{3}
∴ 1729 can be expressed as sum of two perfect natural cubes.
It is the smallest number to satisfy this condition, and, is known as Ramanujan’s Number.
, where, P, A, T, E, F are digits from to what will be the value of F?
x,y are respective carries.
y + P + E + F E
x + 2A = y E
T + T = x A
⇒ P = 9, A = 8, T = 4, E = 6 and F = 1
∴ F = 1
Sum of 3 numbers = product of 3 numbers. If the numbers are consecutive and natural. Find the triplet having least value, of their sum.
xyz = x + y + z …(i) We also know that,
∴ Sum of 3 natural numbers > 3.
∴ From general interpretation,
1 + 2 + 3 = 1 ⇒ 2 ⇒ 3
∴ Required set of natural numbers = (1, 2, 3).
The square of a number is having 5 at its units place and 2 at its tenths place, then the least natural number having these properties are :
∵ Last (unit) place digit = 5
∴ The least natural number whose perfect square is having 5, as its unit place digit will be equal to 5.
∴ 52 = 25
The product 135 × 135 will be equal to :
135 × 135
Trick :
135 × 135 = 25
Multiply the units place 5 and write the product, Add 1 to any one of the remaining digits at tenths place and write the product on tenths place.
13 × 14 = 182
∴ 135 × 135 = 18225
Which of the following number is not a perfect squares ?
∵ Perfect squares should have 1, 4, 9, 6 and 5 as their units place digit.
∴ 1282 is not a perfect square.
26 + 34 × 17 ÷ 4 = 34, which of the two signs should be interchanged to get the desired result?
Interchange of ×, ÷ will produce the desired result.
Which is the least number divisible by 2, 3, 5 and 55 ?
Least number which is divisible by 2, 3, 5 and 55
= LCM 2, 3, 5 and 55
= 66 × 5
= 330
What is the square number just greater than 60, which can be expressed as a sum of two successive triangular numbers?
60 < Sum of two triangular numbers = perfect square. The smallest number satisfying this condition is 64.
What will be one’s place digit for 9^{201}?
9^{1} = 9
9^{2} = 81
9^{3} = 243 × 3 = 729
9^{4} = 6561
9^{5} = 59049
∴ 9^{4n} = 6561
∴ 9^{200} has 1 as its units place digit.
∴ 9^{201} has 9 as its units place digit.
What is the value of P if P, Q, R are replaced by digits from 1 to 9? PQ × QP = RQPR.
P = 8, Q = 7, R = 6
∴ P = 8
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